Basis Decomposition

LiDAR return intensity provides complementary surface-response cues for robotic perception and state estimation, yet many simulation pipelines omit it or reproduce it using reconstruction methods that require real intensity supervision and per-scene optimization. These requirements increase data-collection and fitting costs and limit reuse across simulated scenes. We present NIDAR, a feed-forward framework that synthesizes dense intensity-like observations from RGB appearance and simulator geometry. NIDAR combines pretrained pseudo-NIR translation, hierarchical intrinsic decomposition, geometry-aware modulation, and source-domain distribution calibration to transfer reflectance-related image cues to simulated point clouds. Its learned components are trained offline using Waymo data; their weights and calibration remain fixed during evaluation on Waymo and nuScenes. Deployment therefore requires neither target-scene intensity labels nor target-scene gradient-based fitting. The reported comparisons show competitive pixel-wise accuracy and favorable structural and perceptual fidelity against the evaluated reconstruction baselines. A controlled pseudo-NIR-versus-RGB diagnostic further shows that the pseudo-NIR prior is most beneficial when used through the paper-aligned reflectance-and-remapping route, rather than as a simple direct intensity regressor. We further integrate NIDAR with Unreal Engine 5, Isaac Sim, and a generative LiDAR pipeline. Two intensity-aware SLAM systems evaluated in two simulated indoor scenes suggest potential downstream utility, but do not constitute real-robot validation. NIDAR therefore offers a scalable intensity-synthesis interface for the evaluated settings; cross-wavelength, camera-configuration, embedded, and real-sensor validation remain future work.

Quantized Mixture-of-Experts (MoE) services can hold several pre-materialized instances of one base model, but quantization damage varies sharply across requests and bitwidths. Because instance materialization and replica counts consume memory and require slow reconfiguration, we treat them as upstream provisioning decisions and study routing within a fixed resident pool. Within this fixed-pool boundary, we route each request to maximize modeled throughput under a class-level expected quality-degradation budget and measured instance capacities. To predict this request-specific risk, we introduce FWP (Fragility-Weighted Perplexity), computed from prompt tokens on a reference-instance prefill and calibrated to candidate-instance degradation. Underlying FWP is an exact two-expert affinity--fragility decomposition and a conditional multi-layer top-$k$ expansion whose bias, interaction, route-change, separability, and higher-order terms remain explicit. Using these calibrated risks, a window-level linear program yields a signed reduced-reward score that is KKT-consistent with the LP optimum under optimal prices and primal-feasible tie allocation. On 88 extended Qwen prompts, complete W2, W3, and W4 instances quantizing all 6,144 expert blocks incur mean $\Delta$NLL of $0.9437$, $0.1832$, and $0.0513$. Under the same population and $\tau=0.1513$, FWP allocation reaches a $1.284\times$ offline model-based multiplier versus $1.253\times$ for request-agnostic mixing and $1.000\times$ for static W4, an incremental $2.5\%$ relative FWP gain.

Transformer-based language models are known to sometimes generalize to sequences longer than seen during training, but we lack a precise characterization of which tasks admit length generalization. It is not even known which regular languages transformers length-generalize on -- and this is a foundational class of languages. Our contributions are to establish the first complete characterization of which regular languages transformers length-generalize on and provide a decision algorithm running in polynomial time in the size of the language's syntactic monoid. These results rely on an effective characterization of the regular languages in C-RASP, a recently-established formalism that expresses which languages transformers length-generalize on. This characterization is challenging because classical tools like Krohn-Rhodes decomposition theory for finite semigroups are insufficient for C-RASP. Firstly, the basic building blocks of Krohn-Rhodes theory -- flip-flop and simple groups -- are not expressible in C-RASP. Secondly, the basic building block of C-RASP (unbounded counting) is not expressible by the finite semigroups of Krohn-Rhodes theory. Thus, length generalization on regular languages is controlled by an algebraic property that is invisible to classical finite decomposition theory. We generalize classical decomposition theory from finite semigroups to the infinite additive group on the integers, allowing us to characterize C-RASP in terms of iterated wreath products of the integers and derive a provable polynomial-time decision algorithm for regular language membership. Experiments across a broad test suite of regular languages confirm that our theory captures transformers' length-generalization behavior more accurately than existing classifications.

Symmetric non-negative matrix factorization (SymNMF) recovers latent group structure from a dependence matrix, but its dense, quadratic-memory objective has confined prior work to moderate sizes. We present a large-scale GPU study of seven algorithm families (over 30 configurations) on absolute Pearson correlation and tail pairwise dependence matrices from Extreme Value Theory, two proxies for empirical risk-factor estimation on large portfolios. A trace-identity reformulation eliminates all $n \times n$ intermediates, so a single GPU reaches $n \approx 10^5$ and multi-node distribution scales to $n = 10^6$ and beyond. Under a two-phase protocol, eleven methods converge at moderate scale; six remain efficient enough at $n = 10^5$ (five AdaGrad-family plus ADMM), and five AdaGrad-family methods still converge at $n = 10^6$: AdaGrad, RMSprop, and three we introduce (Piecewise AdaGrad, Row-Stochastic SVRG, Block-SVRG AdaptGrow). At $n = 10^6$ the fastest solver tracks the matrix spectrum: Block-SVRG AdaptGrow wins on the flat, ill-conditioned tail-dependence spectrum, where its lower per-iteration cost decides a long factorization, and full-batch AdaGrad wins on the dominant-low-rank correlation spectrum, where the run is short. We also benchmark spherical K-means as a hard-label baseline: cheaper when angular cluster structure is present, yet provably degenerate once the matrix collapses toward a single common factor, where the soft factorization remains necessary.

Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage--Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.

We demonstrate that AI-driven program synthesis can autonomously discover fundamental strategies for decomposing unitary matrices in photonic networks. By extending DreamCoder to complex-valued linear algebra, the system generates decomposition programs achieving the minimal $N(N-1)/2$ Mach-Zehnder interferometers, distinct from both Reck and Clements architectures. Learned programs encode dimension-agnostic invariants: strategies discovered for $5 \times 5$ matrices generalize to higher dimensions such as $64 \times 64$. The discovered programs encode interpretable, dimension-agnostic construction rules. These rules generalize across matrix sizes without retraining, demonstrating that autonomous program synthesis can serve as a scalable paradigm for algorithm discovery and the automated design of universal unitary operators. Beyond universal decompositions, the system automatically exploits matrix structure to reduce the interferometer count below the universal theoretical bound. For instance, for Householder matrices, it discovers a dimension-independent rule that requires only $2N-3$ MZIs. This achieves linear, rather than quadratic, scaling and generalizes to arbitrary $N$ without retraining. For matrices obtained from the singular value decomposition of sparse matrices, reductions generally increase with sparsity, reaching up to 38% fewer MZIs than the universal theoretical bound $N(N-1)/2$ at 95% sparsity. These MZI reductions translate directly into practical hardware benefits for scalable photonic implementations. Taken together, the system functions as a single unified engine that discovers both universal decomposition rules and matrix-specific optimizations, without being provided with the structural or analytical properties of the input matrices.

Intrinsic decomposition which expresses image colors as the product of diffuse albedo and shading, possibly augmented with view-dependent residuals has a long history in image editing as it enables the modification of object colors and textures without altering lighting. We extend intrinsic decomposition to radiance fields represented with Gaussian splatting by proposing solutions to three key aspects of such decomposition. First, we describe how to model the intrinsic decomposition as independent sets of Gaussian primitives, which allows each set to adapt to the characteristics of the layer it represents. Second, we present an optimization procedure guided by data-driven predictions to disentangle multi-view photographs of a scene into the aforementioned intrinsic sets. Finally, we provide an editing workflow where users modify the texture of planar surfaces simply by modifying the albedo of that surface in one image. Capturing this edit within the intrinsic radiance field allows re-rendering of the edited scene with plausible lighting under arbitrary viewpoints.

Interpretable machine learning requires models that are accurate and structurally faithful to the data. Existing explainability methods rely heavily on additive representations (e.g., Generalized Additive Models (GAMs), SHapley Additive exPlanations (SHAP), functional ANOVA), which can suffer from signal cancellation and off-support extrapolation in the presence of strong interactions. We propose Tensor Separation Learning (TSL), a regression model that learns a sum of rank-1 products of univariate per-feature functions via a stagewise greedy procedure with orthogonal refitting. By enforcing separability, TSL avoids the information loss inherent in additive projections caused by marginalizing higher-order interactions. The learned TSL model can be fully reconstructed from first-order partial dependence functions, up to constant factors. This stage-wise correspondence ensures that the resulting visualizations are faithful to the fitted components. We establish approximation-rate guarantees for functions with bounded mixed $p$-th order partial derivatives and demonstrate that TSL competes with black-box models on regression benchmarks.

Mueller matrix imaging provides rich, physically meaningful contrast for biomedical tissue analysis, but supervised learning is hindered by scarce dense annotations and strong domain shifts across specimens and acquisition settings. We introduce MuellerPT, a physics guided pre-training approach that learns transferable dense representations by predicting Lu-Chipman decomposition maps from per-pixel 4x4 Mueller matrices. To scale pre-training, we collected a new large Multispectral Animal Polarimetric Organ dataset (MAP-Org). The pre-trained encoder is adapted with a segmentation head for grey vs. white matter segmentation in lamb brain. A classification head is used for colorectal cancer vs. non-cancer classification. Both segmentation and classification are evaluated across few-shot learning scenarios. In segmentation, MuellerPT improves label efficiency and cross specimen transfer compared to models without pre-training, achieving an absolute DICE gain of over 20% compared to the baseline trained from scratch when using 5% of the training data. In classification, MuellerPT also enhances label efficiency, improving overall accuracy by 8% compared to the baseline when using 1% of the training data. We demonstrate MuellerPT's robustness to domain shift with a qualitative evaluation of its predicted Lu-Chipman maps on an ex vivo human oesophagus sample. These results suggest that predicting Lu-Chipman decomposition is an effective and practical pretext task for robust biomedical inference from Mueller polarimetry and can pave the way for future work on label efficient Mueller imaging.

The functional ANOVA, or Hoeffding decomposition, provides a principled framework for interpretability by decomposing a model prediction into main effects and higher-order interactions. For independent inputs, this classical decomposition is explicit. It is closely connected to SHAP values, generalized additive models, and orthogonal polynomial expansions, and therefore constitutes a fundamental tool for additive explainability. In the more general and realistic dependent setting, however, obtaining a tractable representation and estimating the decomposition from data remain challenging. In this work, we address this problem for continuous inputs. By combining Hilbert space methods with the generalized functional ANOVA, we build an explicit decomposition Riesz Basis allowing to easily compute the decomposition. Our formulation recovers the classical independent case and its associated orthogonal decomposition. Building on this representation, we propose a simple but mighty algorithm to estimate the decomposition from a data sample in a model-agnostic setting and we compare it empirically with several state-of-the-art explanation methods, demonstrating the power of the approach.

The alignment of Large Language Models (LLMs) utilizes Reinforcement Learning from AI Feedback (RLAIF) for non-verifiable domains such as long-form question answering and open-ended instruction following. These domains often rely on LLM based auto-raters to provide granular, multi-tier discrete rewards (e.g., 1-10 rubrics) that are inherently stochastic due to prompt sensitivity and sampling randomness. We empirically verify the stochasticity of auto-raters that can propagate and corrupt standard advantage estimators like GRPO and MaxRL, as a noisy reward samples can skew normalization statistics and degrade the global learning signal. Empirically, sampling more rewards and taking majority voting may reduce the noise and improve performance, but this approach is computationally expensive. To address this bottleneck, we introduce $\textbf{O}$rdinal $\textbf{D}$ecomposition for $\textbf{R}$obust $\textbf{P}$olicy $\textbf{O}$ptimization ($\textbf{ODRPO}$), a framework that structurally isolates evaluation noise by decomposing discrete rewards into a sequence of ordinal binary indicators. By independently computing and accumulating advantages across these progressively challenging success thresholds, ODRPO prevents outlier evaluations from corrupting the global update while establishing an implicit, variance-aware learning curriculum. Empirically, ODRPO achieves robust performance on Qwen2.5-7B and Qwen3-4B models, outperforming baselines with relative improvements of upto 14.8% on FACTS-grounding-v2 and 7.5% on Alpaca-Evals. Critically, these gains are achieved with negligible training-time overhead, as ODRPO requires no additional compute per step compared to standard estimators. Supported by theoretical analysis confirming its optimization stability, ODRPO provides a scalable and robust framework for aligning models within the noisy, discrete evaluation landscape of modern RLAIF.

Knowledge graph embedding (KGE) models typically represent each relation as an operator on entity embeddings. In this work, we identify three structural axioms that any principled relation operator must satisfy, linearity, trace preservation, and complete positivity, and show that they characterize a Kraus channel structure via the Kraus representation theorem. The completeness constraint defining this family is equivalent to these axioms, providing a principled foundation rather than an externally imposed condition. Under this formulation, most existing operator-based KGE models are recoverable as special cases with Kraus rank $κ= 1$ under specific embedding choices. We further generalize this characterization to arbitrary metric geometries by introducing \mbox{w-Kraus} channels, which satisfy completeness by construction within their respective spaces. Building on this theory, we propose \textsc{KrausKGE}, a principled KGE model that naturally handles $1$-to-$N$ and $N$-to-$N$ relations, supports $k$-hop reasoning without requiring explicit path encoders, and eliminates the need for norm constraints on entity embeddings. Additionally, our framework yields the first theoretically grounded per-relation complexity measure in the KGE literature, with a provable lower bound in terms of the empirical relation matrix rank. Empirical evaluation demonstrates that \textsc{KrausKGE} consistently outperforms strong baselines on $N$-to-$N$ relations, with performance gains that increase monotonically with relation fan-out, in alignment with theoretical predictions.

We study solving large-scale fixed-point equation $x^\star=\bar F(x^\star)$ with decomposition. Standard strict decomposition assigns each agent a disjoint block and evaluates updates using only owned coordinates. For most operators, however, a block update may depend on variables outside the block. Truncating these dependencies by strict decomposition changes the mean operator and creates structural bias that cannot be removed by more samples, smaller stepsizes, or additional consensus. We therefore propose Core-Halo decomposition, which separates write ownership from read-only evaluation context: each agent updates its own core and reads from an overlapping halo. By aligning the Core-Halo decomposition with the block-dependence structure of $\bar F$, the original fixed-point problem can be implemented faithfully in a decentralized multi-agent system. We further characterize the fundamental obstruction faced by strict decomposition through a Bellman closure condition and a blockwise bias lower bound, showing that local-only updates can alter the original fixed-point operator. Finally, we conduct extensive experiments across a range of application settings, and demonstrate that Core-Halo achieves near-centralized performance while retaining the parallelism benefits of decentralization.

Estimating free energy differences quantifies thermodynamic preferences in molecular interactions, which is central to chemistry and drug discovery. Despite fruitful progress, existing methods still face key limitations: classical computational approaches remain prohibitively expensive due to their reliance on extensive molecular dynamics simulations, while deep learning-based methods are constrained by either less-expressive generative models or input dimensions tied to a specific system, resulting in negligible generalization. To address these challenges, we propose CARD, a generative framework that employs a novel radix-based decomposition to bijectively convert 3D coordinates into mixed discrete-continuous sequences, enabling coarse-to-fine autoregressive modeling with enhanced expressiveness. Notably, the model corresponds to a distribution with zero free energy, serving as a proposal for absolute free energy computation of arbitrary systems without relying on alchemical pathways. Experiments across diverse tasks demonstrate that CARD matches the accuracy of classical computational methods on unseen systems with diverse topologies, while achieving an approximately 40-fold speedup in inference.

Implicit neural representations (INRs) are increasingly being used as tools to map coordinates to signals, encompassing applications from neural fields to texture compression, shape representations, and beyond. Most INR methods are based on using high-dimensional projections of the initial coordinates through encoders such as grid or positional encoding. Nevertheless, positional encoding is often insufficient and grids, as we show in this paper, require high resolution for being able to learn. In this paper, we demonstrate that positional encoding can be used not only as a high-dimensional embedding but also decomposed as a series of meaningful points. We propose the Positional Encoding Projected Sampling, where we treat the projection of the original coordinate at each frequency as a point of interest. We describe the motion of each point with respect to the frequencies and show that it follows a unique pattern. Finally, we use the unique motion of each point as a basis decomposition for doing learned positional encoding using grids. We prove, using three competitive applications; image representation, texture compression, and signed distance function; that the proposed approach outperforms the current state of the art methods, and often requires 25% less parameters for equivalent reconstruction error or rendering.

In uncertainty quantification, evaluating sensitivity measures under specific conditions (i.e., conditional Sobol' indices) is essential for systems with parameterized responses, such as spatial fields or varying operating conditions. Traditional approaches often rely on point-wise modeling, which is computationally expensive and may lack consistency across the parameter space. This paper demonstrates that for a pre-trained global Polynomial Chaos Expansion (PCE) model, the analytical conditional Sobol' indices are inherently embedded within its basis functions. By leveraging the tensor-product property of PCE bases, we reformulate the global expansion into a set of analytical coefficient fields that depend on the conditioning variables. Based on the preservation of orthogonality under conditional probability measures, we derive closed-form expressions for conditional variances and Sobol' indices. This framework bypasses the need for repetitive modeling or additional sampling, transforming conditional sensitivity analysis into a purely algebraic post-processing step. Numerical benchmarks indicate that the proposed method ensures physical coherence and offers superior numerical robustness and computational efficiency compared to conventional point-wise approaches.

Adaptive Neuro-Fuzzy Inference System (ANFIS) was designed to combine the learning capabilities of neural network with the reasoning transparency of fuzzy logic. However, conventional ANFIS architectures suffer from structural complexity, where the product-based inference mechanism causes an exponential explosion of rules in high-dimensional spaces. We herein propose the Kolmogorov-Arnold Neuro-Fuzzy Inference System (KANFIS), a compact neuro-symbolic architecture that unifies fuzzy reasoning with additive function decomposition. KANFIS employs an additive aggregation mechanism, under which both model parameters and rule complexity scale linearly with input dimensionality rather than exponentially. Furthermore, KANFIS is compatible with both Type-1 (T1) and Interval Type-2 (IT2) fuzzy logic systems, enabling explicit modeling of uncertainty and ambiguity in fuzzy representations. By using sparse masking mechanisms, KANFIS generates compact and structured rule sets, resulting in an intrinsically interpretable model with clear rule semantics and transparent inference processes. Empirical results demonstrate that KANFIS achieves competitive performance against representative neural and neuro-fuzzy baselines.

Attention is a core operation in large language models (LLMs). We present BD Attention (BDA), a lossless algorithmic reformulation of attention. BDA is enabled by a simple matrix identity from Basis Decomposition (BD), which restructures multi-head projections into a compact form while preserving exact outputs. Unlike I/O-aware system optimizations such as FlashAttention, BDA provides a mathematically guaranteed acceleration that is architecture-agnostic. On DeepSeek-V2-Lite (16B, FP16), BDA requires only 4s of offline preparation with no retraining required and, on modern GPUs, achieves 34% faster key/value projections and 25% smaller weights, while increasing perplexity (PPL) by just 0.02% (FP16) or 0.0004% (FP32), a negligible effect on model performance. These results position BDA as a theoretically exact method for lossless attention acceleration that is complementary to existing engineering-level optimizations. Our code is available at https://github.com/abcbdf/basis-decomposition-official.