Orthogonality

Orthogonality in a LoRA factor does not by itself specify what the composed update protects: the answer depends on the task-start state, the parameterization, and the realized optimizer displacement. We formalize this question through Bilinear Optimization Divergence (BOD), an anchor-relative diagnostic of effective-update response on selected historical features. The finite-step analysis distinguishes two cases. In a shared adapter, protecting the routing displacement leaves a learned-anchor residual through the changing companion factor. In a fresh zero-output block, a feasible routing state can protect the composed update while both current factors remain trainable. These conditions yield Semi-Frozen Orthogonal Routing (SFOR) for shared adapters and current-block hard protection for cumulative O-LoRA; Weight Residual Projection (WRP) enforces the required displacement after the optimizer step. Controlled two-task traces verify the predicted residual paths, reducing normalized historical response from 19.12% to 0.005% in the shared family and from 7.72% to 0.002% in the cumulative family. Four-task experiments on Qwen3-8B characterize the resulting trade-offs: SFOR improves backward transfer (BWT) from -2.47 to -0.86 with nearly unchanged average accuracy (AA), while O-LoRA hard protection improves three-order mean AA from 80.27% to 81.30% and forgetting measure (FM) from 2.20 to 0.43. Component controls also show that stricter feasibility need not improve final task performance. Together, the analysis and evidence provide an architecture-conditioned account of which constraint to enforce, how to enforce it, and how to interpret its empirical value.

Audio deepfake detectors need to transfer to languages absent from training, as multilingual speech synthesis outpaces labeled anti-spoofing resources. While detectors increasingly rely on self-supervised speech models (S3Ms), these backbones encode language-dependent structure that confounds spoof cues. We address this confound through language orthogonalization, a target-free ridge map that removes S3M variation projected onto continuous language-identification (LID) embeddings. Across six languages, six S3M backbones, and all Leave-N-Out settings, it consistently reduces EER across unseen languages. Cross-lingual EER correlates with LID-space distance, where orthogonalization yields larger gains for more distant transfers.

Detector response nonuniformity introduces systematic projection errors and ring artifacts in photon-counting detector computed tomography (PCD-CT). In measured PCD-CT data, residual stripe amplitudes vary slowly with projection angle, which fixed-bias models cannot adequately capture. We propose TRACE, a two-stage unsupervised sinogram decomposition method for estimating and correcting these response-related errors. TRACE represents stripes as a fixed bias plus low-order discrete cosine transform (DCT) components, using a small number of coefficients to describe angular variations at each detector element. A learnable analysis--synthesis architecture represents the ideal projections, while two-stage optimization separates them from fixed and then dynamic stripes. An angular-gradient soft orthogonality constraint suppresses correlated variations within the shared DCT gradient subspace, reducing the leakage of object structures into the artifact estimate. All parameters are optimized directly on the measured sinogram without paired training data. Experiments on measured QRM mouse phantom and porcine trotter data show that TRACE suppresses ring artifacts and improves image uniformity while preserving edge sharpness, soft-tissue texture, and trabecular detail.

We develop the Sparse Orthogonal Regression Technique (SORT), a sparse spectral framework for learning orthonormal-basis expansions from noisy and irregularly sampled data. SORT estimates expansion coefficients directly from observations using L1-regularized regression, avoiding explicit quadrature or analytic inner-product evaluation. The central application is data-driven discovery of ordinary differential equations: vector fields are represented in chosen orthogonal bases and learned as sparse coefficient expansions. This provides a complementary route to symbolic regression, grammar-based discovery, and SINDy-style sparse identification by first recovering a compact spectral representation, which can later guide searches for simpler analytic forms. Across the dynamical-system experiments, SORT matches or improves upon library-based sparse-regression baselines when the basis is well adapted to the problem, and shows more stable degradation under sparse sampling, noisy derivative estimates, and representation mismatch. Specific examples illustrate why this representation is useful: if a finite library misses the problem-specific nonlinearity, the resulting model can fail. SORT is not immune to mismatch, but it shifts the problem away from brittle selection among generic terms to basis design adapted to the problem domain. The experiments also show that dominant low-order coefficients persist as model order increases, supporting order-consistent model growth. Beyond equation discovery, the same learned expansion supports nonlinear approximation and estimation of complex, high-dimensional integrals by coefficient readout. Overall, SORT provides a reusable intermediate representation for system identification, approximation, and integration, while making basis design an explicit part of the scientific modeling problem.

Rotation-based post-training quantisation commonly applies an orthogonal transform across an entire attention head to reduce outlier-induced error. RoPE instead partitions each head into two-dimensional frequency pairs, raising the question of whether a transform respecting this decomposition can improve on full-head mixing. Prior work has established the per-pair rotations that commute with RoPE. We state the converse result that, for distinct frequencies, no other single-head orthogonal map commutes with RoPE. For the head-shared parameterisation used in our experiments, we then derive the rotation angle that minimises the larger channel variance under a pooled-covariance, position-averaged surrogate and verify that the implementation attains its analytic minimum. The evaluated head-shared pairwise configuration does not improve accuracy in the tested dynamic W4A4KV4 setting. Across four checkpoints, replacing the full-head Hadamard with this configuration increases perplexity at both short and long context lengths. Composing the pairwise rotation with the Hadamard satisfies the selected $\pm0.05$-PPL interval criterion under the default estimator. Estimating the shared angle from K alone improves pairwise-only on every checkpoint but does not close its gap to full-head mixing. The analytic objective controls a position-averaged second moment of a pooled calibration covariance, whereas the dynamic quantiser sets its step from a tokenwise group range. The pairwise transform also has only two-channel mixing support. Along a controlled interpolation from two-channel to full-head mixing, K range, relative quantisation error, and perplexity degradation decrease as support increases. These results show that optimality for a structured surrogate need not reduce quantisation error when the surrogate and mixing support are misaligned with the quantiser's scale-setting statistic.

Distance-based reliability estimation assumes that a representation's geometry reflects its trustworthiness, yet this assumption is rarely tested under training interventions that reshape geometry directly. We audit this assumption under domain-adversarial representation learning using a disentanglement dose-response ladder. Three checkpoint families share the same architecture and a 16-dimensional representation, differing only in orthogonality strength (lambda = 0, 1, 5). Representation geometry changed substantially with disentanglement strength: the condition number shifted by two orders of magnitude (Kendall tau = 0.84, exact p = 2.8e-5). This change was not accompanied by improved reliability estimation: Mahalanobis-distance AUROC (ISIC-test vs. PAD-UFES) remained flat and below chance (about 0.40) at every level, with no significant association with any of five geometry metrics tested. The same failure was observed for cosine-to-centroid and pooled k-nearest-neighbor scorers, plus three non-distance-based scorers: an energy-based confidence score, Virtual-Logit Matching, and a kernel density estimator. Seven of eight scorers converged on the same result; the energy-based score showed an isolated upward trend that we report but do not treat as evidence against the overall pattern. A supervised probe with no access to the training objective recovered domain membership from the identical embeddings at 0.72-0.81 AUROC across every level, showing that the relevant information was not absent from the representation. These findings indicate that classification performance alone can overlook whether information in a learned representation is organized in a form that downstream reliability estimators can use. Information can remain decodable while becoming largely inaccessible to non-probing reliability estimators.

Learning from heterogeneous representations is usually reduced to feature concatenation, which erases which representation produced an error. We instead algebraize the residual: a representation is a typed object that owns both a coordinate system and the residual it leaves unresolved, and learning is an ordered composition of operators that preserve or deliberately erase that type. Fold realizes the objects as point-in-time conditional-mean fields on 10x10 rank grids. FPRC-PQ realizes the algebra as relax-aggregate-close: each field is relaxed by a correction fitted to its own residual in its own coordinates; corrected fields meet at a fixed mean that is the sole identity-erasure boundary; and a shared learner closes only the aggregate's fresh residual. The composition telescopes exactly into representation, local residual estimate, and residual-of-residual estimate. Its aggregate is a learned control-variate interface with population variance reduction, while refitting the closer along perturbations of the backbone yields first-order coupled-path mean orthogonality. As an analytical extension, a reflective rumination operator reads the displacement of a global reconstruction from the aggregate anchor, reflects it, and fixes its gain by a unique orthogonal projection rather than return-tuned grid search. On 3.67M Chinese A-share stock-day observations (2023-2026) under a frozen point-in-time protocol, the evaluated base algebra raises net-of-cost return from 13.52% to 19.10% and Sharpe from 1.42 to 2.09. Matched-capacity, unified-residual, identity-free two-stage, and pairwise-only controls all trail it. The gain is therefore not explained by more features or more trees, but by making residual ownership and composition explicit while representation identity is still available.

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce \textit{LieStoNet}, an end-to-end, \emph{template-free} framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics. Code is available at \href{https://github.com/sumit-sinha-seas/LieStoNet_Final.git}{this link}.

Binarizing a polynomial Kolmogorov--Arnold Network (KAN) not only changes parameter precision, but also alters the function space available to each layer. When activations are restricted to ${-1,+1}$, all even powers reduce to $1$ and all odd powers reduce to $x$, causing the elementwise polynomial basis to collapse to constant and first-order responses. We refer to this structural failure as Spatial Orthogonality Collapse. Our proposed BiKAN addresses this critical issue by augmenting each binary KAN layer with selected degree-2 Walsh characters. Fixed circular channel rolls generate pairwise parities, and learned binary projections mix them using the same XNOR--popcount operations as the remaining W1A1 paths. This restores explicit pairwise coordinates without learned routing or multiplier-based feature generation. Experiments on CIFAR-10 confirms that removing parity reduces accuracy by $1.23$ points over five paired seeds ($p=0.003$), the gain increases as width decreases, and accuracy improves monotonically as more parity planes are added. At an equal $\sim$11.9M-parameter budget, parity outperforms conventional widening by $3.09$ points ($p<10^{-4}$). At W1A1, BiKAN reaches $99.48\%$, $84.38\%$, and $55.81\%$ on MNIST, CIFAR-10, and CIFAR-100, respectively. Post-route Zynq-7020 FPGA results show that the repair remains hardware-efficient; the convolutional design cuts DSP usage from 164 to 72 and estimated compute-core latency from 401 to 54.8 ms, while the power-of-two-aware dense design achieves zero-DSP inference with a 0.03-point accuracy loss. The BiKAN implementation is available at https://github.com/OSU-STARLAB/BiKAN.

We introduce Orthogonal-Edge (OrEdge), a lightweight framework for real-time anomaly detection in multi-modal distributed software systems. Unlike existing approaches that rely on computationally expensive attention- and graph-based architectures, OrEdge leverages orthogonal-domain temporal representations to achieve accurate anomaly detection with substantially lower computational complexity and model size. It jointly analyzes heterogeneous monitoring data, including logs, metrics, and traces, to identify abnormal software behavior, capture temporal dependencies, and reduce redundancy across observability signals. At its core, OrEdge incorporates OrEdgeCore, a lightweight orthogonal-domain reconstruction module that captures recurring temporal patterns while suppressing transient variations. Evaluated on three real-world microservice datasets (MSDS, SN, and TT), OrEdge achieves competitive detection performance while reducing the reconstruction model size to at most 9.6K parameters, compared with 20K--143K parameters in existing methods. This compact design enables efficient deployment on resource-constrained edge devices: on Raspberry Pi platforms, OrEdge achieves sub-second inference and reduces inference latency by over an order of magnitude compared with existing approaches. Extensive ablation studies, sensitivity analyses, orthogonal basis evaluations, and qualitative case studies further validate the effectiveness of each design component. Overall, OrEdge demonstrates that orthogonal-domain temporal modeling provides an effective alternative to computationally intensive attention- and graph-based architectures, achieving a favorable balance between detection accuracy and computational efficiency for real-time multi-modal anomaly detection in edge environments. The code is available at https://github.com/theamrzaki/MicroService_Twin_Original.

Multimodal large language models (MLLMs) have enabled long-form video understanding at a scale that was not previously possible. However, the density of relevant content decreases sharply as video sequence length increases, and exposing the model to more irrelevant content measurably reduces its accuracy. In this paper, we address the problem of maximizing query-relevant information in a frame subset selected at inference time, without training. FORGE (Frame Orthogonality in Relevance Geometry) is a model-agnostic method that induces a query-conditioned geometry on a pretrained multimodal embedding space, unifying relevance and diversity into a single objective. In this space, frames that cover independent query-relevant directions are far apart, and selecting the subset of maximum information captures diverse query-relevant content within the budget. Experiments on Video-MME and LongVideoBench at budgets of 16, 32, and 64 frames show that FORGE improves the unified keyframe selection score by 11.0-15.3 points over the strongest training-free baseline and up to doubles keyframe recall (0.415 vs. 0.204 at K=64 on Video-MME). The gains extend to question answering, where accuracy improves in every evaluated setting across eight open-source MLLMs spanning 4B to 32B parameters, by up to 8.7 points over uniform sampling and 5.2 points over the strongest baseline. Our findings suggest that aligning the embedding space with the query's high-dimensional structure is a promising direction for inference-time video understanding.

A Last-Layer Ensemble (LLE), $K$ linear units on one shared frozen feature map, is an efficient single-pass approach to the disagreement-based epistemic uncertainty for out-of-distribution (OOD) detection. Its weakness is that members share the backbone gradient and can converge toward the same function, collapsing the inter-member diversity the signal depends on. Whether last-layer diversity can be restored, and what mitigates the collapse, is an open question. The weight-orthonormality defining Orthonormal Certificates (OC), the weight-orthonormal special case of the LLE, is only an indirect correction; it decorrelates the weights of the members, not their predictions. Here, we instead target the collapse directly in function space, with a Covariance Last-Layer Ensemble (cov-LLE) that places a direct covariance penalty on member activations. Cov-LLE restores the function-space diversity that weight-orthonormality cannot, and at matched $K$ recovers much of the diversity and calibration of a deep ensemble at $1\times$ backbone cost (in-distribution prediction variance $0.05\!\to\!9.3$ vs. $22.1$ ($\times10^{-3}$), and ECE $0.135\!\to\!0.090$ vs. $0.035$, for a $K\times$-cost deep ensemble), at no cost to accuracy. Viewing OC as a last-layer ensemble also organizes detectors into a two-axis taxonomy (by how their units are trained and how their outputs are scored) and exposes the OC score as a magnitude, motivating a scale-invariant, label-free direction score that repairs its near-OOD failure, adding $+0.16$ to $+0.18$ ROC AUC on every backbone.

The Discrete Fourier Transform, the Discrete Cosine Transform, and their block-wise variants underpin most deployed image and video codecs. Their effectiveness rests on three properties: they run in near-linear time (linear up to a polylogarithmic factor), they are exactly invertible, and they carry few to no parameters. In this work, we generalize these bases to isometric multilinear bases, allowing a small number of extra parameters, polylogarithmic in the image size, while preserving all three properties. Given an image dataset, we develop a systematic framework that searches this family for the basis compressing the dataset most effectively: the basis is parameterized as an isometric tensor network, inspired by quantum many-body theory, and trained with Riemannian optimization on the manifold of unitary matrices. Across natural photographs and line drawings, the trained bases consistently improve on their fixed, non-parametric counterparts. On Quick Draw line-drawing compression, they store images in roughly $20\%$ fewer bytes than JPEG's $8 \times 8$ block cosine transform at the same reconstruction quality.

Weights and biases are normally optimized as separate parameter tensors, yet they do not represent separate functions when the input to an affine layer has nonzero mean. For an affine map $z=Wx+b$ with input mean $μ$, a weight update contains a sample-independent displacement $ΔWμ$ that is functionally indistinguishable from a bias update. We call this hidden contribution \emph{boundary motion} and decompose each update into a centered, sample-varying \emph{shape} component and a shared \emph{boundary} component. On a four-layer Transformer trained from scratch on IMDb, the bias-like term $g_bμ^\top$ has a median norm equal to 0.664 of the raw weight-gradient norm across affine layers and training checkpoints. More strikingly, the median ratio $\norm{ΔWμ}/\norm{Δb}$ is 134.7, while $\norm{ΔWμ}/\norm{Δb+ΔWμ}$ is 0.994. Thus, under AdamW, the observed boundary motion is almost entirely realized through the weight matrix rather than the explicit bias. We implement a diagnostic optimizer, Shape--Boundary Orthogonal AdamW (SBO-AdamW), that optimizes $g_W-g_bμ^\top$ and $g_b$ with independent Adam states and compensates the weight-induced boundary displacement. In a single-seed experiment, SBO-AdamW raises validation accuracy from 81.68% to 85.81% and validation-selected test accuracy from 78.73% to 82.73%, with the best validation checkpoint occurring at step 800 instead of step 3000. However, the moving-batch-center compensation produces severe bias-coordinate drift and strongly reduces boundary energy. The present evidence therefore supports hidden boundary motion as an important optimization mechanism, but it does not yet establish a final general-purpose optimizer. A stable centered-affine parameterization is identified as the required next step.

For scale-invariant deep networks, Hyperball-style optimizers have shown strong performance in large-scale training by fixing the norms of matrix-valued parameters and normalizing updates. However, the source of their advantage remains unclear. Starting from the angular displacement between consecutive parameter states, we derive an angular effective learning rate that accounts for the parameter-update angle, parameter norm, and update norm. We also show that the conventional norm-based measure is a special case under parameter-update orthogonality. We then decompose optimizer updates into radial and tangential components and analyze how radial updates affect one-step angular displacement. Under the training configurations considered, numerical results show that the radial component has only a limited direct effect on the angular effective learning rate. It therefore cannot explain why MuonH converges more slowly than MuonWD early in training but overtakes it later. To further isolate the underlying mechanism, we devise a heuristic experiment that modifies only the learning-rate schedule so that the dynamics of each optimizer reproduce those of the other. The results suggest that their main difference stems from the evolution of the effective step size rather than an intrinsically superior update direction induced by Hyperball. Our pretraining experiments further show that more aggressive learning-rate decay can accelerate MuonH early in training but may impair its later performance. Thus, maintaining a constant angular velocity does not eliminate the learning-rate-scheduling problem; careful scheduling remains essential to realizing the potential of Hyperball-style optimizers. Our code is publicly available at https://github.com/mangocrazz/hyperball-may-not-be-a-free-lunch.

Fine-tuning has been widely used to adapt large language models (LLMs) for domain-specific tasks. Parameter efficient fine-tuning (PEFT) methods such as low-rank adaptation (LoRA) are frequently used to reduce computational costs. PortLLM is a training-free and data-free scheme used to adapt LLMs after continual pretraining. Although the initial PortLLM results show that LoRA patches exhibit short-term temporal portability, the long-term performance of PortLLM across several updates of continual pretraining remains underexplored. Furthermore, the intriguing effectiveness of PortLLM is not well understood from a theoretical standpoint. We address these two open questions by (1) performing an extensive empirical study of the long-term temporal portability of PortLLM patches across 10 continual pretraining steps using base models Mistral, Gemma, and Qwen; and (2) offering two theoretical analyses to explain our observation that the simple PortLLM method achieves competitive performance. We find empirically that the portability persists across longer time duration, indicating that repeated fine-tuning is not required when the base model is periodically updated. We find theoretically that near-orthogonality of high-dimensional vectors is a key justification for temporal portability. Our analyses also demonstrate a geometric perspective of the loss landscape in facilitating the theoretical comparison of different adaptation options.

Comparing phylogenetic tree topologies is essential for understanding epidemic dynamics, yet biologically meaningful distances such as the Subtree Prune and Regraft (SPR) distance are NP-hard to compute and intractable on large datasets. We investigate whether a Graph Neural Network (GNN) can approximate SPR distances in near-constant time per comparison after training. Our contributions are fourfold. First, we build and publicly release a dataset of 864 phylogenetic trees inferred with UPGMA and Neighbor-Joining over four bacterial species, spanning up to 9{,}500 isolates, together with 388 labelled tree pairs. Second, we establish a reproducible pre-processing pipeline including midpoint re-rooting, which reduces tree depth and supplies the rooting required for exact distance computation and for the model's root-based features. Third, we validate the supervision target: on small trees, where exact SPR is tractable, the unrooted phangorn::SPR.dist heuristic correlates almost perfectly with the exact rooted distance computed by rspr (Pearson $0.98$--$0.99$), making it an excellent monotonic surrogate. Lastly, we train a Siamese Graph Isomorphism Network (GIN) regressor. In-distribution, i.e., held-out trees from the same species and size range as training, it explains roughly 87--90% of the variance ($R^2 \approx 0.87$ on a held-out split; $0.90 \pm 0.19$ under stratified cross-validation), with about four times lower error than a mean-predictor baseline, and shows partial transfer to unseen species ($R^2 \approx 0.37$). Its main limitation is extrapolation to trees larger than those seen in training, where accuracy collapses. The released dataset and the validated heuristic versus exact relationship provide a reproducible basis for scaling learned SPR approximation.

Time series forecasting requires models to capture diverse, often mutually exclusive, temporal dynamics, from smooth trend continuation to nonstationary drift and strict phase-aligned recurrence. While recent deep learning models have improved accuracy, they typically force these diverse patterns through a single computational backbone governed by fixed algorithmic inductive biases (e.g., self-attention or spectral filtering). This single-mechanism approach often struggles with the profound heterogeneity of real-world series, where different variables and forecast horizons necessitate fundamentally different predictive treatments. To address this, we propose GatedLinear: a lightweight framework that frames forecasting as the adaptive routing of complementary linear bases. GatedLinear leverages a pool of three specialized mechanisms: a global trend-seasonal basis for smooth projection, a difference-based incremental basis for nonstationary drift, and a phase-aligned recurrence basis for explicit cyclic reuse. To dynamically orchestrate these distinct behaviors, we introduce a Tri-Factorized Fusion Gate that disentangles routing decisions into channel-specific preferences, horizon-aware offsets, and phase-indexed biases derived from known future time marks. This design allows the model to perform highly granular, point-wise soft routing across different predictive regimes without stacking computationally heavy neural modules. Experiments on standard benchmarks show that our method achieves state-of-the-art or highly competitive accuracy against recent complex foundational models, while offering explicitly interpretable routing patterns and operating with a substantially smaller parameter footprint.

Audio-language models compress a speech encoder's output through a Querying Transformer (Q-Former) connector before feeding it to a large language model. We identify two failures in this compression. The connector's output vectors collapse to a single direction, and different speakers produce nearly indistinguishable outputs, with paralinguistic cues such as speaker identity, gender, and prosody lost along the way. Our method, ORCA, reverses this collapse by splitting the queries into groups whose outputs are constrained to point in different directions. On SAKURA multi-hop reasoning, ORCA gains 26.4 points over an identically trained 4B baseline, reaching 75.2% (vs. 49.0% for the 8B Audio Flamingo-3). At the connector level, the same change cuts query redundancy by 12x and raises cross-speaker variance by 75x.

Principal Component Analysis or PCA-like properties (orthogonality, variance ranking) are seldom realized in deep autoencoder architectures. In this work, we present ODIN (Orthogonal Dendritic Intrinsic Network), a novel autoencoder architecture that recovers PCA-like latent structure in a fully non-linear regime. By incorporating a set of geometric constraints directly into the training objective, ODIN encourages latent dimensions to be mutually orthogonal and ordered by explained variance, mirroring the interpretable decomposition of PCA while retaining the expressive power of deep networks. We provide theoretical grounding for these constraints and demonstrate their compatibility with standard encoder-decoder frameworks. We also establish empirical results for both synthetic and real world datasets, establishing a principled path toward interpretable, structured feature learning and dimensionality reduction.

Orthogonal and Stiefel layers give neural weights exact spectral control, but they also impose a strong modeling constraint: all represented singular values are fixed at one. Many settings that benefit from an orthonormal basis still need direction-dependent attenuation or amplification. We introduce ManifoldFlow, a minimal relaxation of a fixed-spectrum Stiefel layer that keeps the basis on the Stiefel manifold while learning a bounded positive spectrum through W = Q S^{1/2}, with Q^T Q = I and S positive definite. Since W^T W = S, the eigenvalues of S are exactly the squared singular values of the realized weight, making eigenvalue clipping a direct singular-value control mechanism. Across paired sequence, tabular, and image experiments, the learnable SPD spectrum improves the fixed-spectrum Stiefel counterpart in the reported settings where the Stiefel prior is useful, with the largest gains in recurrent language-model projections. Boundary cases in convolutional classifier heads clarify the intended scope: ManifoldFlow is not a universal dense-layer replacement, but a spectrum-learnable Stiefel relaxation for settings where an orthonormal basis is a useful prior. When the basis should be orthonormal, its spectrum need not be frozen. Code available at https://github.com/Hik289/manifold_flow

Orthogonalizing the mLSTM memory matrix at read time with five differentiable Newton-Schulz iterations improves noisy associative recall. We replicate this effect and investigate its mechanism. Training on MAD noisy recall exhibits a long chance-level plateau followed by a sharp increase in accuracy. The orthogonalized read improves conditioning during this plateau and can be removed after escape. Ablations support three findings. First, the benefit requires a self-consistent read and gradient: an exact recursive least-squares read (the Mesa layer) yields a similar benefit, while straight-through variants, delta-rule writes, frozen random keys, and Frobenius normalization show no improvement over baseline. Second, across a learning-rate x task-difficulty grid, orthogonalization multiplies escape hazard roughly six-fold, with no detectable dependence on difficulty, and widens the range of learning rates that produce successful runs. Third, adding orthogonalization at inference leaves chance-level failures unresolved, while removing it gradually after escape yields standard mLSTMs at near-perfect accuracy. Schedule changes alone recover much of the reported gain. A batch-size x learning-rate analysis separates the effects of per-step learning rate and gradient noise on escape hazard (elasticities +3.0 and -1.65, respectively), linking the original vocab-96 result to its large-batch training regime. Direct decoding of the memory state recovers roughly half of the associations in behaviorally failed models, indicating a readout-learning limitation despite substantial stored information. These results show that fixed-budget recall benchmarks are sensitive to trainability and provide a tractable setting for investigating abrupt behavioral transitions through measurements of internal representations.

We study the symmetric bouncing of a point robot within orthogonally-joined rectangles with equal width, which we refer to as pipes. We provide an exhaustive case analysis of every trajectory pattern inside a single rectangular pipe segment, identifying the conditions under which the robot exits. We then extend the analysis to L-shaped pipes and, more generally, to linear chains of $k$ orthogonally connected pipe segments. We prove exit guarantees for the special angle $α= π/4$. Furthermore, these results extend to pipes with curved joints.

Parameter-Efficient Fine-Tuning (PEFT) commonly adapts pretrained weights through low-rank updates, and recent methods further exploit the singular value decomposition (SVD) of the base weight for initialization or subspace selection. However, these methods do not explicitly preserve the coupled geometry between the pretrained left and right singular bases. Motivated by recent minimum-perturbation theory, which shows that stable finetuning follows a coherent SVD rotation in which a single orthogonal $Q$ acts on both the left singular basis $U_0$ and the right singular basis $V_0$, we prove a per-slice analogue: each row slice of $W_0$ can be adapted by a shared orthogonal rotation $Q_i$ on its left basis $U_i$ and right basis $V_i$ together with a diagonal spectrum shift. We implement this form as CORA (Coherent Orthogonal Rotation Adaptation), which applies per-slice orthogonal rotations and a per-layer diagonal scale to the rank-$r$ SVD truncation of $W_0$. CORA uses $\tfrac{1}{2}m(r{-}1)$ trainable parameters per linear layer, about $4{\times}$ fewer than LoRA at the same rank. CORA outperforms LoRA, DoRA, PiSSA, and MiLoRA on commonsense reasoning and code generation while using about $8{\times}$ fewer parameters.

Low-rank adaptation (LoRA) and its variants enable parameter-efficient fine-tuning of large language models under the supervised fine-tuning (SFT) paradigm. However, their efficacy and behavior under Reinforcement learning with verifiable rewards (RLVR) are less well understood. In particular, two structurally initialized LoRA variants, PiSSA and MiLoRA, which outperform standard LoRA under SFT, can underperform standard LoRA under RLVR and may even exhibit training instability. These observations suggest that how to initialize the low-rank matrices in RLVR remains unclear. In this work, we develop a theoretical analysis of LoRA in RLVR, showing that orthonormal initialization achieves the minimal gap between LoRA outcome and that of full fine-tuning. Guided by this insight, we propose geometry-preserving orthonormal initialization for low-rank adaptation in RLVR, leading to two new variants, RLPO and RLMO. Experiments on mathematical reasoning benchmarks show that the proposed orthonormal initialization stabilizes RLVR training and outperforms standard LoRA, contrasting with PiSSA and MiLoRA. Finally, our unified analysis for LoRA initialization also explains why PiSSA and MiLoRA can underperform in RLVR, which may be of independent interest. Code and checkpoints are publicly available at https://github.com/Richard-ZZZ/geometry-preserving-orthonormal-init-rlvr.

While post-training backdoor detection and trigger inversion schemes have been developed for AIs used e.g. for images, there is a paucity of such methods for LLMs. First, the LLM input space is discrete, with up to 150,000^k k-tuples to consider with k the token-length of a putative trigger. Second, one must blacklist tokens typical of the putative target response (class) of an attack, as such tokens may give false detection signals. However, a comprehensive blacklist is not available, in general, for a given domain. We develop a highly effective detection and inversion framework for LLMs treated as classifiers. Central to our approach is class subspace orthogonalization (CSO), a novel plug-and-play paradigm for backdoor detection that serves two fundamental roles when applied to LLMs: i) it enhances both sensitivity and specificity of a baseline detector; ii) it provides a form of implicit blacklisting, as it penalizes against inclusion, in a candidate trigger, of tokens that induce signal perturbations "in the direction of" the putative target class of an attack. One version of our detector performs continuous optimization in token embedding space, while a companion trigger-inversion and detection method performs greedy accretion in discrete token space. Our methods give both strong detection performance and accurate inversion of ground-truth triggers on several LLM classification domains, and for several different LLM architectures.

Deep learning, which in general relies on voluminous amounts of training data, is vulnerable to data poisoning attacks, including error-generic attacks and backdoors (Trojans). In this work, we propose a new data poisoning attack we dub a latent class attack. Here, all poisoned examples are from a class that is novel (unknown) for the given classification domain and are mislabeled to one of the known classes (the target class) of the domain, so that the model learns to recognize the novel class as a sub-class of the target class. Such attacks could be used e.g. to defeat AI-based access control systems, or could cause a "foe" to be classified as a "friend". We also propose a post-training defense to detect this attack, without any access to the training set. This detection approach builds on "class subspace orthogonalization" (CSO), a plug-and-play paradigm demonstrated to improve existing backdoor detectors. Here, CSO is used to seek an input (a putative unknown class instance) whose internal representation is not aligned with any of the known classes, and yet which is classified with confidence to one of these classes. Finally, specific to image classification domains, we propose a method for visualizing the estimated unknown class instance, providing explainability to our latent class detections.

Unified fashion generation integrates tasks like virtual try-on and garment reconstruction into a single model to reduce task-specific adaptation costs. However, naive parameter sharing across semantically distinct tasks induces negative transfer through severe inter-task gradient conflict. We propose OrthoTryOn, a unified framework mitigating this interference within a shared Low-Rank Adaptation (LoRA) module. Its Orthogonal Subspace Projection (OSP) applies task-specific orthogonal rotations to bottleneck features, mapping them into decorrelated coordinate frames. To address residual semantic coupling at inference time, we further propose Fisher-guided Negative Guidance (FNG), a parameter-free strategy that utilizes diagonal Fisher information to quantify inter-task sensitivity overlap and explicitly repels generation trajectories from the most confusable task via Classifier-Free Guidance. Extensive experiments demonstrate that OrthoTryOn avoids the severe performance degradation typical of naive unified training and even surpasses independently trained task-specific models, achieving state-of-the-art results across multiple benchmarks while generalizing robustly across diverse diffusion backbones. Code is available at https://github.com/NJU-PCALab/OrthoTryOn.

Knowledge editing aims to efficiently update factual information in Large Language Models (LLMs) without full retraining. However, existing methods still suffer from performance degradation in batch knowledge editing. We identify that semantic representation entanglement, such as overlapping concepts and shared syntactic patterns, accumulates interference in the representation space and reduces editing precision. To bridge this gap, in this paper, we propose Orthogonal Representation Editing (ORE), which performs edits in the hidden representation space of LLMs by constructing a general semantic subspace and enforcing orthogonal constraints on edit vectors, effectively decoupling semantic entanglement. Furthermore, we introduce a gated non-linear representation head to enable adaptive learning of editing locations and precise control over knowledge injection. Extensive experiments show that ORE outperforms existing methods and achieves superior performance in cross-lingual knowledge editing scenarios. We release our code at https://github.com/YVVH/ORE.

Language models are widely used in assistant settings, where controlling behavioral attributes is often essential. Activation steering modifies hidden-state representations at inference time, providing a lightweight, training-free mechanism that can be toggled at runtime. Existing methods, however, have focused primarily on steering a single attribute at a time. When multiple attributes must be controlled simultaneously, naive summation of per-attribute steering vectors suffers from norm imbalance and directional cancellation, while classifier-based approaches require retraining whenever the attribute set changes. We introduce ORBIT (Orthogonal Rotation-Based Intervention Technique), a training-free extension of rotation-based steering to the multi-attribute setting. Our method constructs a joint subspace from per-attribute steering planes via singular value decomposition and applies a single norm-preserving rotation within that subspace toward a combined target direction. Adaptive per-token gating identifies which attributes need correction at each position, and an optional additive boost strengthens attributes with weak initial projection. We also introduce TraitFactory, a new multi-attribute benchmark that focuses on behavioral tendencies rather than surface-level style. We evaluate ORBIT on TraitFactory and ToneBank across three models (Llama-3.2-3B, Qwen-2.5-7B, Llama-3.1-8B) while steering multiple attributes simultaneously, showing that it achieves stronger and more balanced multi-attribute steering than existing training-free baselines while better preserving output coherence.

This paper explores unsupervised disentangled representation learning from a functional perspective. We define latent concepts as factors that influence observations through locally orthogonal directions, formalized as an orthogonality constraint on the Jacobian of the generative mapping. We prove that this condition yields identifiability of general nonlinear generative models, without requiring statistical independence or causal assumptions, provided the latent domain admits all combinations of factor values. Experiments with orthogonality-regularized normalizing flows empirically confirm the theory, demonstrate reliable recovery of ground-truth factors, and shed light on the success of VAEs. These findings challenge the prevailing impossibility claims for unsupervised disentanglement and provide a principled alternative foundation.

We introduce a semi-parametric framework for nonlinear system identification, which decouples discrepancy functions from physics-based components. Orthogonal Gaussian process regression balances sparse parameter selection (the white box) with discrepancy learning (the black box) to produce interpretable models from incomplete physics.

While sequential residual fitting is the bedrock of standard boosting frameworks, it inherently breeds learner redundancy by repeatedly revisiting correlated error components. To address this bottleneck, we propose a shift from residual fitting to \textit{residual orthogonalization} and introduce SCBoost. Our framework tackles redundancy through two complementary mechanisms: Spectral Residual Projection (SRP) and Covariance-Regularized Weighting (CRW). During training, SRP projects each residual target onto the orthogonal complement of the historical prediction subspace, forcing successive learners to capture only novel empirical innovations. During aggregation, CRW optimizes ensemble weights on a validation set with an explicit covariance penalty to mitigate remaining correlations. Theoretically, we provide a finite-sample geometric characterization proving that SRP yields an exact additive residual-energy decomposition. Furthermore, under an isotropic-noise assumption, we rigorously establish the conditions under which this projection improves the effective Signal-to-Noise Ratio. Extensive experiments across ten benchmark datasets demonstrate that SCBoost delivers strong out-of-the-box performance, particularly in accuracy and F1 score. This work reinterprets boosting through a geometric lens, suggesting that explicit redundancy control is a principled and necessary step toward more efficient ensemble architectures.

Muon is an optimizer that computes updates using the polar factor of the momentum matrix and has shown strong empirical performance across a range of training settings. A key component of Muon is the Newton-Schulz iteration used to compute this polar factor. Although this avoids the cost of an exact singular value decomposition, it remains expensive in practice because it is applied at every optimization step. At the same time, the momentum matrix changes smoothly over training, suggesting strong temporal correlation in the corresponding polar factors. In this paper, we exploit this structure and propose CacheMuon, a temporal preconditioning method that reuses information from previous optimization steps to approximate the polar factor at the current step. This reduces redundant orthogonalization computation across iterations. We analyze CacheMuon as an inexact Muon update, with error controlled by fresh-solver error and cache staleness. Empirically, CacheMuon provides a controllable quality-efficiency frontier: conservative thresholds closely match fresh Muon on language-model and vision training while reducing orthogonalization FLOPs, whereas more aggressive thresholds yield larger arithmetic savings at the cost of modest validation-quality degradation.

Modern deep learning architectures are increasingly multi-task and multi-modal, using a pretrained foundation model combined with task-specific, fine-tuned models. Empirically, exploiting similarity across different problems, instead of solving them individually, can significantly improve overall performance. While the generalization and sample complexity properties of multitask learning have been widely studied, the parametric complexity of joint approximation in comparison to separate approximation remains less well understood. The question is particularly relevant in modern deep learning, where models are increasingly required to satisfy structural constraints such as equivariance, conservation laws, or orthogonality. We prove lower and upper bounds on the description-length for separate and joint approximation classes, respectively, in uniform norm. We build a class of orthogonal functions by composing a shared hard feature, realized by a Rademacher-Haar wavelet series, with Sawtooth-Walsh readouts to enforce orthogonality of output coordinates. The dyadic tree structure of the Rademacher-Haar wavelet concentrates the approximation hardness in the common feature component, while the readouts act as task-specific heads. Using an information-theoretic framework, we obtain a sharp gap between the optimal approximation rates achievable by joint and separate coding. Finally, we realize this separation in a neural network model using Heaviside activations via reduction to triangle-wave approximation. Our results show that even under an orthogonality constraint joint approximation requires strictly fewer bits in compositional architectures, provided the tasks share a latent hard feature. This provides theoretical insight into the description-length-efficiency of compositional multi-output architectures and clarifies how neural networks can retain expressivity under geometric constraints.

Backpropagation (BP) is widely viewed as biologically implausible, in part because it requires feedback weights to be the transpose of forward weights for error propagation. Interestingly, when training a network with fixed random feedback weights to circumvent this issue, learning aligns the forward weights with the feedback weights, leading the backpropagated error signal to become an approximation of the standard gradient used by BP. This process, called Feedback Alignment (FA), occurs in MLPs and very shallow CNNs but does not scale well to deeper architectures. In this work, we first investigated differences between BP and FA models, trained on CIFAR10, specifically focusing on the effective rank of the signal. We found that the FA error has a considerably lower rank and hence is constrained to a lower-dimensional subspace compared to BP, limiting exploration of the parameter space. Motivated by this observation, we evaluated two mechanisms for increasing the effective dimensionality of FA: Muon, an optimiser that orthogonalises weight updates; and hidden activity normalisation, which promotes activation orthogonality. Across larger architectures and benchmarks, we find that these methods consistently improve over FA baselines, for example, on CIFAR100 with a Resnet-18, accuracy increases by 9 percentage points. Our results identify low-dimensional gradient dynamics as a key obstacle to scaling FA and suggest that inducing higher-dimensional update geometry is a promising route toward scaling alternatives to backpropagation.

Low-Rank Adaptation (LoRA) enables efficient federated fine-tuning of segmentation foundation models for medical imaging. However, most federated LoRA methods adopt a uniform aggregation rule, which breaks under the encoder-decoder asymmetry in medical segmentation: the encoder is dominated by appearance shifts, while the decoder is dominated by supervision variations. This mismatch entangles shared anatomy with site-specific biases and harms generalization. To address this, we propose Inverse Asymmetric Tuning (IAT). IAT aligns adaptation with heterogeneity sources by personalizing module-specific components in the encoder to absorb appearance shifts and in the decoder to accommodate site-dependent supervision, while retaining a shared pathway for transferable consensus. However, structural separation alone is insufficient under LoRA's bilinear parameterization, where multiplicative coupling can still cause site-specific updates to leak into the shared direction. We therefore introduce a Subspace Orthogonality Regularizer that penalizes shared-local collinearity in the effective update space, mitigating leakage without extra communication. Experiments show consistent improvements over strong federated LoRA and parameter-efficient FL baselines.

This methodological study analyzes the effects of collinearity, effective dimensionality, and cluster stability in a 2023 study of US airline profit cycles from 1995 to 2020 by Renold et al., which uses k-means clustering, principal component analysis, and system dynamic modelling.We replicate their clustering experiment in three spaces -- the original 7-dim. raw-variable space, a 3-dim. PC score space, and a 4-dim. PC score space using their dataset. We show that the six-cluster taxonomy is geometrically robust: k-means in 3-PC space produces bit-for-bit identical cluster assignments relative to 7D raw space. As a nonlinearity check we apply kernel PCA under six kernels spanning three families plus a linear baseline. The kernels confirm an intrinsically linear manifold with no detectable curvature. The silhouette criterion reveals that the dataset structurally supports only three clusters, not six. Collinearity in the raw 7D space suppresses the silhouette signal. A kernel ridge regression check confirms no nonlinear accuracy gain over linear ridge once the COVID19 year is excluded. Together, these results argue for clustering on PC scores rather than raw variables in collinearity-prone panel data.

We study the symmetric multi-type orthogonal non-negative matrix tri-factorization problem, where several symmetric non-negative matrices are simultaneously approximated by factors of the form $GS_{i}G^{\top}$, with a shared non-negative and orthogonal factor $G$. This model is motivated by clustering and network analysis, where non-negativity improves interpretability and orthogonality gives a natural assignment-type structure to the latent factor. Since the resulting optimization problem is highly non-convex, we develop two heuristic algorithms for computing high-quality local solutions. The first one is a fixed point method derived from the Karush-Kuhn-Tucker conditions after adding a penalty term for the orthogonality constraint. The second one is a three-stage ADAM-based method that combines non-negativity-preserving optimization, orthogonalization, and restricted ADAM refinement on the feasible set. We evaluate both methods on synthetic data, including noisy instances, and on citation network benchmarks. The synthetic experiments show that both algorithms recover factorizations close to the optimum and remain stable under noise. On real networks, the learned embeddings are competitive with or better than standard baselines such as SVD, node2vec, and classical link prediction heuristics in link prediction, node clustering, and node classification tasks.

Muon has recently demonstrated strong empirical performance in large language model training, but the theoretical role of momentum in Muon remains unclear. Existing analyses of Muon either remove momentum to study spectral updates in isolation, or retain momentum without explaining why it improves empirical performance. Our work bridges this gap by showing momentum in Muon acts as a spectral filter. Under a structured signal-plus-perturbation gradient model, we prove that momentum suppresses perturbations while preserving the dominant signal, thereby enlarging the spectral gap between them. This enlarged gap stabilizes the singular subspaces of the matrix passed to Muon's orthogonalization step, making the resulting update more reliable. We further show that applying momentum before orthogonalization achieves provably stronger alignment with the signal component of the gradient than either reversing this order or simply removing momentum. Experiments across diverse tasks, including LLM pretraining, support our theoretical analysis. More broadly, our theory offers a starting point for understanding the benefits of momentum in other matrix-based optimizers.

Model dimension ($d_{model}$) is a fundamental hyperparameter in transformer language models, yet its role in setting the geometric limits of feature representation remains under-explored. Grounded in the Linear Representation and Superposition Hypotheses - which propose that models encode features as near-orthogonal directions in latent space - we develop a framework for estimating how many such directions a model can support. We first establish the embedding matrix as a measurable proxy for near-orthogonality constraints across the latent space: the boundary between meaningful token relationships and incidental similarity in the pairwise cosine similarity distribution gives a concrete estimate of the model's accepted deviation $\varepsilon$ from perfect orthogonality. Applying this metric across dozens of open-source models reveals two classes: models with high $\varepsilon$ whose embeddings lack near-orthogonal structure, and models with low $\varepsilon$ that maintain it. We then show that the standard Johnson-Lindenstrauss lemma greatly underestimates the packing efficiency of trained representations, and derive an adjusted capacity formula in which the number of near-orthogonal directions depends on the ratio of vectors to dimensions ($k/d$) rather than the raw count - a single modification that cuts prediction error by two orders of magnitude with no extra parameters. Combining these results, we define representational capacity as an upper bound on the number of distinguishable directions available for features and embeddings in a model's latent space. Capacity is exponentially sensitive to $\varepsilon$, and larger models favor tighter orthogonality constraints over maximizing raw capacity - a pattern compatible with several explanations (a stability-capacity trade-off, a ceiling on usable concepts, or confounds with model scale) that we leave to future work.

We study finite-sample generalization for a client-sampled distributed optimization scheme with matrix-valued parameters and orthogonalized momentum updates. The central quantity is the gap between the population and empirical objectives at the returned model when only a subset of clients participates in each round. Under independent heterogeneous client data, unequal local sample counts, and fixed aggregation weights, we derive a finite-round upper-tail guarantee from a coupled-neighbor stability recursion and a weighted concentration step. The bound keeps the client-selection counts through the amplification factor $Y_i(\mathcal C)$; in the uniform full-participation full-batch regime, it yields $\widetilde{\mathcal O}(n^{-1}+n^{-1/2})$ scaling whenever the horizon-dependent amplification terms are controlled. The matrix-orthogonalization rule is required to be Lipschitz along paired trajectories, a condition satisfied by regularized polar-type maps and normalized finite-step Newton--Schulz orthogonalizers. For the unregularized matrix sign, the same argument requires coupled spectral separation, whereas Gaussian smoothing gives a finite-round smoothed variant. A one-dimensional counterexample shows why a gap, smoothing, or regularity condition is necessary.

Muon optimizers improve neural-network training by replacing ill-conditioned momentum updates with approximately semi-orthogonal updates. This motivates a practical question: how much orthogonalization does Muon actually require? We study this question using a relaxed cubic Newton--Schulz schedule derived directly for Muon's low precision singular value band. The resulting five-step cubic construction uses ten dominant matrix multiplications, compared with fifteen for five quintic Newton--Schulz iterations. The cubic schedule is not intended as a more accurate polar solver; instead, it is a principled low-cost variant that lets us probe the relation between polar accuracy, spectral shaping, and training quality. Across synthetic diagnostics, NanoGPT ablations, and training experiments on hybrid MoE/Mamba models, we find that training quality is not governed monotonically by polar-decomposition accuracy: truncated Polar Express, Muon-Jordan, cubic Newton--Schulz, and an explicit FP32 SVD polar factor can reach nearly indistinguishable final loss on GPT-2 Small, and cubic5 matches the Muon-Jordan quintic update within about $10^{-3}$ validation loss on hybrid MoE/Mamba models with one billion to four billion parameters. These results support cubic5 as a practical low-cost Muon orthogonalization variant, with empirical evidence of training-quality parity in the settings tested.

Supervised Contrastive Learning (SupCon) has achieved strong performance by explicitly modeling pairwise relationships among samples. However, existing SupCon-based methods suffer from two key limitations: negative-sample dilution induced by the standard InfoNCE loss, and feature-space entanglement caused by the lack of explicit constraints separating category-relevant (common) and category-irrelevant (style) features. These limitations reduce feature discriminability and generalization ability. To address these issues, we propose OSCS-SupCon (Orthogonal Sigmoid-based Common and Style Supervised Contrastive Learning), a unified framework that combines a sigmoid-based pairwise contrastive objective with explicit orthogonality constraints. Specifically, we introduce a sigmoid-based contrastive loss with two learnable parameters, temperature and bias, which adaptively modulate pairwise decision boundaries and alleviate negative-sample dilution. Furthermore, we enforce orthogonality between common and style feature subspaces via a linear projection with ReLU nonlinearity, thereby reducing feature overlap and improving disentanglement of style-irrelevant representations. Extensive experiments on six benchmark datasets demonstrate that OSCS-SupCon consistently outperforms state-of-the-art supervised contrastive learning methods across multiple backbone architectures. In particular, on the fine-grained CUB200-2011 dataset with a ResNet-18 backbone, the proposed method achieves a 3.4% improvement in classification accuracy over CS-SupCon, highlighting its robustness and generalization capability. Ablation studies further confirm the effectiveness of each component.

The massive scaling of Large Language Models (LLMs) has made pretraining increasingly cost-prohibitive. While low-rank representation and orthonormal weight matrices could in principle reduce parameter counts and computational overhead, most existing methods rely on static rank selection and do not enforce weight orthonormality due to high computational cost. This paper introduces TSVD, a framework that maintains low rank and strict orthonormality throughout the training process. It utilizes a spectral energy-based heuristic for adaptive rank selection, and a caching mechanisms to maintain orthonormality. Theoretical analysis justifies the advantage of the approach in pretraining dynamics and experiments across various model scales demonstrate that it is effective empirically. TSVD matches or exceeds the performance of full-parameter baselines while significantly reducing compute requirements. The approach thus offers a well-founded, practical, and scalable path toward efficient high-performance LLM pretraining.

The successful training of neural networks hinges on the use of first order optimization methods, yet the theoretical characterization of these methods remains incomplete. This is especially true in settings with mild overparameterization. In this work, we study the gradient flow dynamics of two-layer ReLU networks from small initialization with orthogonal training data. We prove the limiting flow converges to a saddle-to-saddle jump process as the initialization scale tends to zero, revealing an incremental learning phenomenon in which a new neuron activates at each saddle. This analysis recovers the known result of Dana et al. (2025, arXiv:2502.16977) that the network interpolates the training data with high probability as soon as $m \gtrsim \log(n)$, where $m$ is the network width and $n$ is the number of training samples. This incremental process characterization also allows us to derive a novel implicit bias result: the learned interpolator has a squared $\ell_2$-norm scaling as $\sqrt{n}$, which is within a constant factor of the minimal $\ell_2$-norm interpolator. More broadly, our work provides the first rigorous proof of an incremental learning process for ReLU networks, whilst suggesting mildly overparameterized networks can converge to interpolating solutions whose complexity is of the same order as that of the optimal interpolator.

The Muon optimizer has recently offered a promising alternative to AdamW for large language model training, leveraging matrix orthogonalization to produce geometry-aware updates. However, like all first-order methods, Muon can become trapped in sharp local minima. In this work, we present MONA, an optimizer that bridges Muon's orthogonalization framework with curvature-aware acceleration. MONA adds an acceleration term directly into Muon's gradient processing pipeline. This term is calculated from the exponential moving average of gradient differences. We provide a detailed convergence analysis for MONA, showing that the acceleration term introduces curvature-sensitive corrections while preserving Muon's spectral-norm regularization. Empirically, MONA achieves better convergence and downstream task performance compared to both Muon and AdamW across three scales of Mixture-of-Experts pretraining, spanning from 1B to 68B parameters, with the largest model trained on 1 trillion tokens. Furthermore, we conduct supervised fine-tuning on the MOE-68B-A3B model and evaluate it on general capability, mathematical reasoning, and code generation benchmarks, where MONA achieves SOTA performance.

The deployment of Large Language Models (LLMs) and Vision Transformers (ViTs) on edge devices is significantly constrained by memory limitations and the critical timing bottlenecks introduced by dense Multiply-Accumulate (MAC) arrays. In the ultra-low bit regime, logarithmic Power-of-Two (PoT) quantization provides a hardware-efficient alternative by replacing MAC operations with bit-shifts. However, the non-uniform exponential lattice is inherently limited by a \textbf{Low Angular Resolution Regime}, a structural flaw that becomes particularly pronounced at sub-4-bit thresholds, leading to a notable degradation of high-dimensional feature manifolds. To address this geometric limitation, we propose Geometric Orthogonal Residual Projection Quantization (GoQuant), an algorithm-hardware co-design framework. By formulating quantization as a dual-basis geometric projection, GoQuant adaptively synthesizes a higher-resolution residual lattice using strictly shift-and-add operations. Furthermore, its analytical solver offers a practical alternative to computationally intensive gradient-based optimization, reducing the full-model calibration time for LLaMA-2-7B to approximately 15 minutes. Extensive evaluations demonstrate GoQuant's applicability across modalities and its hardware efficiency. Under the 3-bit (W3/A16) constraint, it achieves a perplexity of 6.10 on LLaMA-2-7B, comparing favorably to conventional MAC-intensive baselines like AWQ without relying on asymmetric scaling, while maintaining competitive accuracy in 4-bit scenarios. At the silicon level, standard-cell RTL synthesis at a 28nm node indicates that GoQuant effectively mitigates the timing bottlenecks associated with dense multiplier trees. By flattening the combinational logic depth, our parallel shift-and-add datapath reduces the critical path delay to 0.35 ns.

Infinite-dimensional orthonormal basis expansions play a central role in representing and computing with function spaces due to their favorable linear algebraic properties. However, common bases such as Fourier or wavelets are fixed and do not adapt to the structure of a given problem or dataset. In this paper, we aim to represent these bases with neural networks and optimize them. Our key idea is that any target infinite-dimensional orthonormal basis can be viewed either as a point on the Lie manifold of the orthogonal group, or equivalently, as the endpoint of a continuous path on that manifold that connects a reference basis, e.g. Fourier, to that target. Paths on the Lie manifold satisfy ordinary differential equations (ODEs) governed by skew-adjoint integral operators. Using neural networks to define finite-rank generators of such ODEs allows us to parameterize and optimize orthonormal bases in function space. While relying on finite-rank generators to model infinite operators might seem restrictive, we prove a universality result: even with a rank-2 generator, the integrated solutions of the ODE are dense in the orthogonal group under the appropriate operator topology. In other words, for any target orthonormal basis, there exists a path originating from a reference basis and driven by finite-rank generators that gets arbitrarily close to that target basis. We demonstrate the flexibility of our framework by transforming the Fourier basis into the principal components of a functional dataset, eigenfunctions of linear operators, or dynamic modes of energy-preserving physical simulations.

Muon has recently emerged as a competitive alternative to AdamW for large-scale pre-training, with orthogonalization via Newton-Schulz (NS) iterations as its core operation. Existing Muon variants apply a uniform NS schedule to all parameter matrices, overlooking possible differences in orthogonalization difficulty and its impact on performance. Through a systematic empirical study, we show that this per-matrix heterogeneity is pervasive and largely determined by matrix geometry, which evolves dynamically across operator types, training stages, and network depths. As a result, uniform NS schedules can lead to uneven orthogonalization quality across the model. Motivated by these findings, we propose Adaptive Muon Orthogonalization (AMO), an observe-then-commit method that measures weight geometry by operator type early in training and then uses these signals to allocate the NS budget for the remainder of training. AMO delivers consistent improvements over uniform-schedule Muon across standard, prolonged, and continual pre-training, surpassing the strongest baseline by +0.76 on Llama3.1-1.4B and +0.51 on Qwen3-1.7B in average downstream performance of 12 evaluation tasks.

Recent healthcare foundation models have achieved strong predictive performance through large scale self supervised learning, yet their latent representations frequently entangle physiologic severity, intervention intensity, observational structure, and institutional workflow into shared embedding directions. While effective for downstream prediction, such representations remain semantically opaque and unstable under contextual shift. We introduce AURORA, Adaptive Uncertainty aware Representations through Orthogonalized Relational Alignment, a new framework for healthcare representation learning based on contextual latent geometry. Rather than optimizing a single unified embedding manifold, AURORA decomposes representations into orthogonal semantic subspaces corresponding to distinct contextual factors and learns relational consistency objectives within each subspace. This induces latent spaces that are both semantically disentangled and geometrically interpretable. Across multiple clinical prediction and retrieval tasks, AURORA consistently outperforms reconstruction, contrastive, and self distillation baselines while substantially improving contextual disentanglement, neighborhood purity, and robustness under institutional distribution shift. Our results suggest that latent geometry itself constitutes an important axis of healthcare foundation model design and that explicitly structuring representation space according to contextual semantics provides a complementary direction beyond conventional predictive compression objectives.

We present a new algorithm for deciding formula entailment in orthologic (a sound approximation of classical logic) that avoids the costly preprocessing phase of prior implementations while retaining the same $\mathcal{O}(n^2(1+|A|))$ worst-case complexity. We then introduce a family of synthetic SAT benchmarks based on the observation that, for any formula $φ$, the equivalence $φ\leftrightarrow \mathrm{NF}_{\mathrm{OL}}(φ)$ is a tautology whose Tseitin encoding yields unsatisfiable instances that are hard for state-of-the-art SAT solvers yet have short orthologic proofs. Applied to EPFL arithmetic circuits, our algorithm solves these instances efficiently while Kissat times out on a significant fraction. Finally, we show that using orthologic normalization as a preprocessing step can improve SAT solving time on some hard problems.

Alignment, the tendency of adjacent weight matrices in deep networks to develop compatible subspace orientations, underlies gradient flow, Neural Collapse, and representation similarity across architectures. Despite extensive empirical documentation, these phenomena have resisted unified theoretical treatment: existing explanations are post-hoc, each fitted to a specific observation with whatever mathematics is at hand. We reverse this direction by deriving the mathematical structure that layerwise alignment inherently demands. Using geometric invariant theory, we prove that alignment geometry has a canonical closed, polystable stratum given by a flag variety, and that subspace intersection dimension is its unique reparameterization-invariant observable, establishing that subspace metrics are not empirical conventions but mathematical necessities. This unified framework yields two dynamical consequences: ridge regularization drives subspace alignment at an exponential rate set by weight decay, whereas nonlinear activations induce a commutator obstruction to exact basis alignment, generically present in nonlinear networks and absent in linear ones. Together these give a geometric explanation of the Level-2/3 hierarchy in Neural Collapse from first principles rather than post-hoc analysis. The commutator magnitude and head subspace overlap further serve as weight-space windows into internal alignment structure, requiring no forward passes. Experiments on multilayer perceptrons, residual networks, and pretrained language models support the proposed diagnostics and delineate their scope.

Steering or intervening on model representations at inference time to correct predictions is essential for AI interpretability and safety, yet existing evaluation protocols are limited to ambiguous language modeling tasks. To address this gap, we introduce SwordBench, a benchmark for steering image representations of vision models across multiple backbones and concept removal tasks. Beyond a unified benchmarking suite, we propose new evaluation notions that uncover the second-order effects of orthogonalization among concept activation vectors for pragmatic steering. Specifically, cross-concept robustness measures the stability of concept detection performance across inputs orthogonalized against alternative concepts, and collateral damage quantifies whether steering inadvertently affects model performance on a downstream task for inputs lacking the bias. We find that although a linear support vector machine exhibits superior separability and orthogonality, it fails to achieve zero collateral damage, often trailing sparse autoencoders. In simpler regimes, both standard baselines and optimization-based methods fail to achieve perfect steering. The source code will be made available soon on GitHub.

Zeroth-order (ZO) optimization has become increasingly popular and important in fine-tuning large language models (LLMs), especially on edge devices due to its ability to adjust the model to local data without the need for memory-intensive back-propagation. Recent works try to reduce ZO variance through low-dimensional subspace search, but subspace restriction alone leaves key optimization geometry under-exploited, motivating additional acceleration. In this work, we focus on the hidden layer training problem in which spectral optimizers like Muon outperform AdamW due to its ability to exploit weak spectral directions by orthogonalization. However, we have discovered that unlike in the first-order setting, full orthogonalization works poorly in the ZO setting since the gradient estimates are highly noisy and unreliable. To address this issue, we propose applying partial spectral orthogonalization to accelerate ZO optimization. To do so, we replace the iconic Newton-Schulz procedure in Muon with the faster, more concentrated power-iteration method so that it only amplifies dominant spectral directions. Furthermore, to improve the efficiency and generalization of the algorithm, we adopted a streaming variant of power-iteration that requires low variance in gradients, which was achieved through constraining our search inside a subspace obtained through the projection of momentum, echoing recent advances. Experiments on LLM fine-tuning show that our method can achieve from 1.5x to 4x the convergence speed of ZO-Muon, the current SOTA algorithm, across SuperGlue datasets in the OPT-13B model. Across different models, we also reach competitive final accuracies with less time in most cases compared with strong ZO baselines such as MeZO, LOZO and ZO-Muon. Code is available at https://github.com/MOFA-LAB/ZO-MOPI.git.

It has been known for a long time that initializing weight matrices to be orthogonal instead of having i.i.d. Gaussian components can improve training performance. This phenomenon can be analyzed using finite-width corrections, where the infinite-width statistics are supplemented by a power series in $1/\mathrm{width}$. In particular, recent empirical results by Day et al. show that the tensors appearing in this treatment stabilize for large depth, as opposed to the tensors of i.i.d.-initialized networks. In this article, we derive explicit layer-wise recursion relations for the tensors appearing in the finite-width expansion of the network statistics in the case of orthogonal initializations. We also provide an extension of recently-introduced Feynman diagrams for the corresponding recursions in the i.i.d.-case which are valid to all orders in $1/\mathrm{width}$. Finally, we show explicitly that the recursions we derive reproduce the stability of the finite-width tensors which was observed for activation functions with vanishing fixed point. This work therefore provides a theoretical explanation for the stability of nonlinear networks of finite width initialized with orthogonal weights, closing a long-standing gap in the literature. We validate our theoretical results experimentally by showing that numerical solutions of our recursion relations and their analytical large-depth expansions agree excellently with Monte-Carlo estimates from network ensembles.

Optimizers that exploit the matrix structure of gradients are central to modern LLM pre-training, with two distinct frontiers: explicit Kronecker-factored preconditioning -- most recently KL-Shampoo, which estimates the preconditioner via KL divergence minimization -- and orthogonalization of the gradient momentum, exemplified by Muon and analyzed as steepest descent under the spectral norm. The two routes are typically developed in isolation. We make a structural observation about KL-Shampoo's Kronecker preconditioners: their eigenvalue spectra exhibit a \emph{spike-and-flat} shape -- a few dominant eigenvalues followed by an approximately uniform tail -- across layers and training stages, holding exactly under a rank-$ρ$ signal-plus-noise gradient model. We exploit this structure by restricting one of KL-Shampoo's Kronecker factors to a parametric family aligned with the spike-and-flat shape: full spectral structure on a tracked $r$-dimensional subspace, single shared eigenvalue across the remaining $n-r$ directions. On these directions, we apply orthogonalization. An identity shows that this orthogonalization recovers the algebraic form of full KL-Shampoo's preconditioner. On four pre-training scales (GPT-2 124M / 350M, LLaMA 134M / 450M), Pro-KLShampoo consistently outperforms KL-Shampoo at every subspace rank we test in validation loss, peak per-GPU memory, and wallclock time to reach each loss level.

We present the E$Δ$-MHC-Geo Transformer, a novel architecture that unifies Manifold-Constrained Hyper-Connections (mHC), Deep Delta Learning (DDL), and the Cayley transform to obtain input-adaptive, unconditionally orthogonal residual connections. Unlike DDL, whose Householder operator is orthogonal only at $β\in \{0,2\}$, our Data-Dependent Cayley rotation $Q(x)=(I+(β/2)A(x))^{-1}(I-(β/2)A(x))$ preserves orthogonality for all $β$ and all inputs. To handle negation, an eigenvalue $-1$ case that Cayley provably excludes, we introduce the E$Δ$-MHC-Geo Hybrid, which combines Cayley rotation with Householder reflection via a learned operator-selection gate $X'=γ(X)Q(X)X+(1-γ(X))H_2(X)X$. A midpoint-collapse regularizer, $4γ(1-γ)$, encourages boundary gate decisions, where each selected component is orthogonal. In matched-parameter comparisons, with approximately 1.79M parameters per model and mean +/- standard deviation over 3 seeds, against four baselines including the concurrent JPmHC, E$Δ$-MHC-Geo achieves the best long-horizon stability, 1.9x over JPmHC and 3.8x over GPT; the best near-$π$ rotation loss, 4.5x over JPmHC on single-plane; strong norm preservation, with 0.001 mean deviation; and 0.96 negation cosine alignment in a diagnostic reflection probe, all with 33% fewer layers. While JPmHC's wider representation excels on pure rotation, its finite Cayley residual mixer excludes an exact $λ=-1$ operator and has no reflection branch, motivating our hybrid approach for accessing both connected components of $O(n)$.

Unified visual tokenization faces a fundamental trade-off between high-fidelity pixel reconstruction (spatial equivariance) and semantic abstraction (conceptual invariance). We attribute this conflict to Manifold Misalignment: naive joint optimization induces opposing gradients, creating a zero-sum game between reconstruction and perception. To address this, we propose MUSE, a framework based on Topological Orthogonality. By treating Structure as an orthogonal bridge, MUSE decouples optimization within Transformers: structural gradients refine attention topology, while semantic gradients update feature values. This turns destructive interference into Mutual Reinforcement. Experiments show that MUSE breaks the trade-off, achieving state-of-the-art generation quality (gFID 3.08) and surpassing its teacher InternViT-300M in linear probing (85.2% vs. 82.5%), demonstrating that structurally aligned reconstruction can enhance semantic perception. Code is available at https://github.com/PanqiYang1/MUSE.

Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have $$\Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} α_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/p},$$ where $c=2$, $1/p+1/p'=1$, and $α_{jk}=\sqrt{\frac{\|f_{j}f_{k}\|_{p/2}}{\|f_{j}\|_{p}\|f_{k}\|_{p}}}$. In the first part of this paper we construct a counterexample showing that this inequality fails for every $p>2$. We then prove that if an estimate of the above form holds, the exponent must satisfy $c\le p'$. Finally, at the critical exponent $c=p'$, we establish the inequality for all integer values $p\ge 2$. In the second part of the paper we obtain a sharp three-function bound $$\Big\|\sum_{j=1}^{3} f_j\Big\|_p \ \le\ \left(1+2Γ^{c(p)}\right)^{1/p'} \Big(\sum_{j=1}^{3} \|f_j\|_p^p\Big)^{1/p},$$ where $p \geq 3$, $c(p) = \frac{2\ln(2)}{(p-2)\ln(3)+2\ln(2)}$ and $Γ=Γ(f_1,f_2,f_3)\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$. The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \frac{6}{5p-4}$ obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.