Residual Learning

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Overview

Residual learning studies how models can build on an existing representation or prediction by adding learned updates, rather than learning the entire transformation from scratch. In deep networks, skip connections and newer approaches such as attention-based routing and multi-stream hyper-connections shape information and gradients across layers; in hybrid models, learned corrections refine physics-based, statistical, or frozen-model predictions. These methods are evaluated in settings including image classification, language modeling, diffusion, time-series forecasting, scientific simulation, and control. A central challenge is determining when residual pathways improve depth, stability, and generalization—and when their complexity, added compute, or poorly transferable corrections instead hurt performance.

Latest papers

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158 papers32 papers in the last four weeks, up 23% on the four weeks before. 0.2% of all new papers.

May 7, 2026

The EΔΔ-MHC-Geo Transformer: Adaptive Geodesic Operations with Guaranteed Orthogonality

We present the EΔΔ-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∈{0,2}β\in \{0,2\}, our Data-Dependent Cayley rotation Q(x)=(I+(β/2)A(x))−1(I−(β/2)A(x))Q(x)=(I+(β/2)A(x))^{-1}(I-(β/2)A(x)) preserves orthogonality for all ββ and all inputs. To handle negation, an eigenvalue −1-1 case that Cayley provably excludes, we introduce the EΔΔ-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))H2(X)XX'=γ(X)Q(X)X+(1-γ(X))H_2(X)X. A midpoint-collapse regularizer, 4γ(1−γ)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, EΔΔ-MHC-Geo achieves the best long-horizon stability, 1.9x over JPmHC and 3.8x over GPT; the best near-ππ 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=−1λ=-1 operator and has no reflection branch, motivating our hybrid approach for accessing both connected components of O(n)O(n).

Arash Shahmansoori

Independent Researcher

TransformerResidual LearningHyper-ConnectionsOrthogonal Representation Learning
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May 5, 2026

SATFormer: Learning When to Use Early Value Residuals in Transformers

Recent Transformer architectures expose later layers to representations computed in earlier layers, motivated by the observation that low-level features can become harder to recover as the residual stream is transformed through depth. One such method adds static value residuals: learned mixing coefficients that expose the first-layer value projection V_1 uniformly across tokens and heads. However, we theorize that the usefulness of V1V_1 is unlikely to be constant across tokens, heads, and contexts, and therefore treat early-representation reuse as a retrieval problem. Here, we introduce Selective Access Transformer (SATFormer), which preserves the first-layer value pathway while controlling access with a context-dependent gate. Across models from 130M to 1.3B parameters, SATFormer improves validation loss over the static value-residual and Transformer baselines at every evaluated scale, and improves average zero-shot accuracy over both at all four scales. Its strongest gains appear on retrieval-intensive benchmarks, where it improves over static value residuals by approximately 1.5 average points, while maintaining near baseline throughput and memory. Mechanistic analyses suggest sparse, depth-dependent, head-specific, and category-sensitive access patterns, supporting the interpretation that SATFormer learns selective reuse of early representations rather than uniform residual copying. Our code is available at https://github.com/skyegunasekaran/satformer.

Skye Gunasekaran, Téa Wright, Rui-Jie Zhu +1

UC Santa Cruz · UC Berkeley

TransformerResidual LearningTransformer Attention
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May 5, 2026

RPBA-Net: An Interpretable Residual Pyramid Bilateral Affine Network for RAW-Domain ISP Enhancement

To address module fragmentation, uninterpretable mappings, and deployment constraints in RAW-domain demosaicing, color correction, and detail enhancement, this paper proposes RPBA-Net, an interpretable residual pyramid bilateral affine network for RAW-domain ISP enhancement. Given packed RAW as input, the method performs residual affine base reconstruction by estimating a base RGB representation and learning identity-guided residual affine corrections, thereby unifying demosaicing and enhancement. It further builds pyramid bilateral affine grids and combines guide-driven autoregressive adaptive slicing with adaptive cross-layer fusion to hierarchically model global tone restoration and local texture enhancement. In addition, smoothness, cross-scale consistency, and magnitude regularization terms are introduced to improve model stability, controllability, and structural interpretability. Extensive experiments demonstrate that RPBA-Net surpasses representative RAW-to-sRGB methods and achieves state-of-the-art performance in reconstruction fidelity and perceptual quality, while maintaining low model complexity and strong deployment potential for mobile and embedded platforms.

Yucheng Xin, Wu Chen, Xiang Chen +7

School of Information Science and Engineering, Shandong Normal University, Jinan, China · National University of Defense Technology, Changsha, China · Nanjing University of Science and Technology, Nanjing, China +2

Image EnhancementResidual LearningDemosaicing
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May 5, 2026

Robust Path Tracking for Vehicles via Continuous-Time Residual Learning: An ICODE-MPPI Approach

Model Predictive Path Integral (MPPI) control is a powerful sampling-based strategy for nonlinear autonomous systems. However, its performance is often bottlenecked by the fidelity of nominal dynamics. We propose ICODE-MPPI, a robust framework that leverages Input Concomitant Neural Ordinary Differential Equations (ICODEs) to learn and compensate for unmodeled residual dynamics. Unlike discrete-time learners, ICODEs maintain physical consistency and temporal continuity during the MPPI prediction horizon. High-fidelity simulations on complex trajectories demonstrate that ICODE-MPPI achieves up to a 69% reduction in cross-tracking error under persistent disturbances compared to standard MPPI control. Furthermore, our analysis confirms that ICODE-MPPI significantly suppresses control chattering, yielding smoother steering commands and superior robust performance.

Shugen Song, Wenjie Mei, Chengyan Zhao

School of Automation, Southeast University, Nanjing 210096, China · School of Automation and the Key Laboratory of MCCSE of the Ministry of Education, Southeast University, Nanjing 210096, China · Graduate School of Computer Science and Systems Engineering, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka-shi, Fukuoka, Japan

Nonlinear MPCResidual LearningAutonomous DrivingRobust ControlMPPI ControlNeural ODEs
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May 4, 2026

Forecasting Medium-Horizon Alzheimer's Disease Progression: Residual Gap-Aware Transformers for 24-Month CDR-SB Change from ADNI Clinical and Biomarker Histories

Medium-horizon Alzheimer's disease progression prediction is difficult because future clinical scores can remain tied to baseline severity, while biomarker histories are irregular and incompletely observed. We develop an anchor-based analysis of 24-month Clinical Dementia Rating Sum of Boxes (CDR-SB) change using harmonized Alzheimer's Disease Neuroimaging Initiative (ADNI) tables. Each labeled sample is anchored at a mild cognitive impairment visit, uses only clinical and biomarker history observed at or before that anchor, and defines the response as CDR-SB at the future visit closest to 24 months within an 18--30 month window minus anchor CDR-SB. The analytic cohort contains 2,600 labeled anchors from 858 participants and 7,276 longitudinal rows. We propose a residual gap-aware transformer that combines a mixed-effects statistical reference with transformer-based residual learning from pre-anchor clinical and biomarker histories. The model uses participant-level random intercepts in the mixed-effects reference, observation-level triplet tokenization for irregular histories, and a learned nonnegative time-gap penalty inside self-attention. We compare the proposed model with a Bayesian-information-criterion-selected linear mixed-effects baseline, GRU-D, and STraTS under repeated participant-level train--test splits. Across five participant-level random seeds, the proposed model achieves the best mean test performance across all reported metrics, reducing MSE by 13.1% and increasing prediction--observation correlation by 26.4% relative to the mixed-effects baseline. It also improves over both GRU-D and STraTS in mean error and correlation. These results show that statistical anchoring and gap-aware residual learning provide a useful structure for medium-horizon Alzheimer's disease progression prediction.

Ran Tong, Tong Wang, Lanruo Wang +1

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080, United States · Department of Plant Science and Landscape Architecture, University of Connecticut, Storrs, CT, United States · Naveen Jindal School of Management, University of Texas at Dallas, Richardson, TX 75080, United States +1

Residual LearningAlzheimer's DiseaseTime Series TransformersIrregular Time-Series ModelingClinical Outcome PredictionAlzheimer's Disease Progression Prediction+1
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May 4, 2026

Geometric and Spectral Alignment for Deep Neural Network I

Deep residual architectures are modeled as products of near-identity Jacobians. This paper proves deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors, emphasizing a normalized top-radial Cartan coordinate and fitted power-law chart. Full-rank factors are mapped from GL(d)\mathrm{GL}(d) to the positive cone by A↦A⊤AA\mapsto A^\top A, then to ordered eigenvalue data. Under Frobenius normalization, exact power-law spectra form a trace-normalized Cartan orbit. This orbit is a Gibbs family on ranks, a Fisher information line, and a Bures--Wasserstein curve with line element d/4d/4 times Fisher information. The main rigidity theorem is a slack-aware margin inequality: interface radial amplitude, non-backtracking slack, and signed residual variation control displacement of the fitted Cartan coordinate. In the exact-chart zero-slack case, a depth-LL budget gives exponent drift of order (log⁡M)/L(\log M)/L; generally, slack and residual increments augment the bound. We separate scalar top-radial from full-Cartan spectral control, which also needs Bures/Hellinger residual variation. We prove approximate-power-law and metric-chart versions, converse lower bounds, Fisher--KL/Bures action estimates, and near-identity expansions for normalized residual chains. Near-identity results verify transport budgets; chart quality remains measurable. Effective rank is a spectral-energy quantile, giving finite-width power-law tail bounds and robust rank-window transition estimates. Empirical static-weight exponent profiles serve as diagnostics; full verification also requires interface budgets, slacks, and residuals for the same operator chain.

Ziran Liu, Wei Wang, Jinhao Wang +5

Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai 200433, China · Futurewei Technologies, Inc., San Jose, CA 95131 · Dept. of Computer Science and Engineering, Santa Clara University, Santa Clara, CA 95050 +2

Residual LearningSpectral Methods
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Apr 27, 2026

Progressive Approximation in Deep Residual Networks: Theory and Validation

The Universal Approximation Theorem (UAT) guarantees universal function approximation but does not explain how residual models distribute approximation across layers. We reframe residual networks as a layer-wise approximation process that builds an approximation trajectory from input to target, and prove the existence of progressive trajectories where error decreases monotonically with depth. It reveals that residual networks can implement structured, step-by-step refinement rather than end-to-end (E2E) black-box mapping. Building on this, we propose Layer-wise Progressive Approximation (LPA), a theoretically grounded training principle that explicitly aligns each layer with its residual target to realize such trajectories. LPA is architecture-agnostic: we observe progressive behavior in residual FNNs, ResNets, and Transformers across tasks including complex surface fitting, image classification, and NLP with LLMs for generation and classification. Crucially, this enables ``train once, use NN models": a single network yields useful predictions at every depth, supporting efficient shallow inference without retraining. Our work unifies approximation theory with practical deep learning, providing a new lens on representation learning and a flexible framework for multi-depth deployment. The source code will be released unpon acceptance at https://(open_upon_acceptance).

Wei Wang, Xiao-Yong Wei, Qing Li

Department of Computing, the Hong Kong Polytechnic University.

Efficient Neural Network InferenceResidual LearningUniversal ApproximationNeural Network Approximation TheoryEarly-Exit Neural Networks
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Apr 26, 2026

Can an MLP Absorb Its Own Skip Connection Exactly?

The benefits usually attributed to skip connections are optimization-theoretic: a smoother loss landscape and better gradient propagation. We ask a representational question instead: given a residual block x -> x + MLP(x), does a residual-free MLP of the same width compute the same function? The answer is no, unconditionally and at every depth, for every activation used in current frontier language models (ReLU^2, ReGLU, SwiGLU, GeGLU). For ungated ReLU and GELU absorption is possible, but only on a set of weights of measure zero. The two families are therefore generically disjoint: removing a skip connection and retraining cannot recover exactly the same function at equal width.

Antonij Mijoski, Marko Karbevski

IRMA, Universit´e de Strasbourg, France · Independent Researcher

Multilayer PerceptronsResidual LearningNeural Network Expressivity
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Apr 20, 2026

Dissipative Latent Residual Physics-Informed Neural Networks for Modeling and Identification of Electromechanical Systems

Accurate dynamical modeling is essential for simulation and control of embodied systems, yet first-principles models of electromechanical systems often fail to capture complex dissipative effects such as joint friction, stray losses, and structural damping. While residual-learning physics-informed neural networks (PINNs) can effectively augment imperfect first-principles models with data-driven components, the residual terms are typically implemented as unconstrained multilayer perceptrons (MLPs), which may inadvertently inject artificial energy into the system. To more faithfully model the dissipative dynamics, we propose DiLaR-PINN, a dissipative latent residual PINN designed to learn unmodeled dissipative effects in a physically consistent manner. Structurally, the residual network operates only on unmeasurable (latent) state components and is parameterized in a skew-dissipative form that guarantees non-increasing energy for any choice of network parameters. To enable stable and data-efficient training under partial measurability of the state, we further develop a recurrent rollout scheme with a curriculum-based sequence length extension strategy. We validate DiLaR-PINN on a real-world helicopter system and compare it against four baselines: a pure physical model (without a residual network), an unstructured residual MLP, a DiLaR variant with a soft dissipativity constraint, and a black-box LSTM. The results demonstrate that DiLaR-PINN more accurately captures dissipative effects and achieves superior long-horizon extrapolation performance.

Youyuan Long, Gokhan Solak, Arash Ajoudani

Human-Robot Interfaces and Interaction Lab, Istituto Italiano di Tecnologia, 16163 Genoa, Italy

Dynamical SystemsResidual LearningNonlinear System IdentificationPhysics-Informed ML
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Apr 18, 2026

Rethinking Cross-Dose PET Denoising: Mitigating Averaging Effects via Residual Noise Learning

Cross-dose denoising for low-dose positron emission tomography (LDPET) has been proposed to address the limited generalization of models trained at a single noise level. However, neural networks trained on a specific dose level often fail to generalize to other dose conditions due to variations in noise magnitude and statistical properties. Conventional "one-size-for-all" models attempt to mitigate this variability but tend to learn averaged representations across noise levels, resulting in degraded performance. In this work, we analyze this limitation and show that standard training formulations implicitly optimize an expectation over heterogeneous noise distributions, causing the network to learn an averaged denoising mapping that cannot accurately model dose-specific noise characteristics. We propose a unified residual noise learning framework that estimates noise directly from low-dose PET images rather than predicting full-dose images. Experiments on large-scale multi-dose PET datasets from two medical centers demonstrate that the proposed method outperforms the "one-size-for-all" model, individual dose-specific U-Net models, and dose-conditioned approaches, achieving improved denoising performance. These results indicate that residual noise learning effectively mitigates the averaging effect and enhances generalization for cross-dose PET denoising.

Yichao Liu, Zongru Shao, Yueyang Teng +1

IWR, Heidelberg University, 69120, Heidelberg, Germany · Silicon Austria Labs, 4040, Linz, Austria · College of Medicine and Biological Information Engineering, Key Laboratory of Intelligent Computing in Medical Image, Ministry of Education, Northeastern University, 110169, Shenyang, China +1

Residual LearningPET/CT ImagingImage Denoising
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Apr 14, 2026

SMART Fine-tuning Factor Augmented Neural Lasso

Fine-tuning is a widely used strategy for adapting pre-trained models to new tasks, yet its methodology and theoretical properties in high-dimensional nonparametric settings with variable selection have not yet been developed. We propose a source-model-augmented residual tuning (SMART) framework, which incorporates the pre-trained source model as an augmented feature into the target learner and estimates only the residual target-specific component. The approach is widely applicable, from parametric and sparse models to neural networks and blackbox machine learning models. We focus on the development of fine-tuning factor-augmented neural Lasso, resulting in SMART-FAN-Lasso. This transfer-learning framework for high-dimensional nonparametric regression with variable selection simultaneously handles covariate and posterior shifts. We use a low-rank factor structure to manage high-dimensional dependent covariates and a residual tuning decomposition in which the target function is expressed as a function of source model and other target-specific variables, thereby reducing the effective complexity of the target task. We derive minimax-optimal excess risk bounds, characterizing the precise conditions, in terms of relative sample sizes and function complexities, under which fine-tuning yields statistical acceleration over single-task learning. Extensive numerical experiments across diverse covariate- and posterior-shift scenarios demonstrate that SMART-FAN-Lasso consistently outperforms standard baselines and achieves near-oracle performance even under severe target sample size constraints, empirically validating the derived rates.

Jinhang Chai, Jianqing Fan, Cheng Gao +1

Department of Operations Research and Financial Engineering, Princeton University

Fine-TuningTransfer LearningNonparametric RegressionFeature SelectionStatistical Learning TheoryResidual Learning
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Apr 13, 2026

Multi-Head Residual-Gated DeepONet for Coherent Nonlinear Wave Dynamics

Coherent nonlinear wave dynamics are often strongly shaped by a compact set of physically meaningful descriptors of the initial state. Traditional neural operators typically treat the input-output mapping as a largely black-box high-dimensional regression problem, without explicitly exploiting this structured physical context. Common feature-integration strategies usually rely on direct concatenation or FiLM-style affine modulation in hidden latent spaces. Here we introduce a different paradigm, loosely inspired by the complementary roles of state evolution and physically meaningful observables in quantum mechanics: the wave field is learned through a standard DeepONet state pathway, while compact physical descriptors follow a parallel conditioning pathway and act as residual modulation factors on the state prediction. Based on this idea, we develop a Multi-Head Residual-Gated DeepONet (MH-RG), which combines a pre-branch residual modulator, a branch residual gate, and a trunk residual gate with a low-rank multi-head mechanism to capture multiple complementary conditioned response patterns without prohibitive parameter growth. We evaluate the framework on representative benchmarks including highly nonlinear conservative wave dynamics and dissipative trapped dynamics and further perform detailed mechanistic analyses of the learned multi-head gating behavior. Compared with feature-augmented baselines, MH-RG DeepONet achieves consistently lower error while better preserving phase coherence and the fidelity of physically relevant dynamical quantities.

Zhiwei Fan, Yiming Pan, Daniel Coca

School of Engineering, Newcastle University, NE1 7RU, United Kingdom · School of Physical Science and Technology, ShanghaiTech University, Shanghai 200031, China

Deep Operator NetworksNeural OperatorsDynamical SystemsResidual Learning
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Apr 2, 2026

Beyond Logit Adjustment: A Residual Decomposition Framework for Long-Tailed Reranking

Long-tailed classification, where a small number of frequent classes dominate many rare ones, remains challenging because models systematically favor frequent classes at inference time. Existing post-hoc methods such as logit adjustment address this by adding a fixed classwise offset to the base-model logits. However, the correction required to restore the relative ranking of two classes need not be constant across inputs, and a fixed offset cannot adapt to such variation. We study this problem through Bayes-optimal reranking on a base-model top-k shortlist. The gap between the optimal score and the base score, the residual correction, decomposes into a classwise component that is constant within each class, and a pairwise component that depends on the input and competing labels. When the residual is purely classwise, a fixed offset suffices to recover the Bayes-optimal ordering. We further show that when the same label pair induces incompatible ordering constraints across contexts, no fixed offset can achieve this recovery. This decomposition leads to testable predictions regarding when pairwise correction can improve performance and when cannot. We develop REPAIR (Reranking via Pairwise residual correction), a lightweight post-hoc reranker that combines a shrinkage-stabilized classwise term with a linear pairwise term driven by competition features on the shortlist. Experiments on nine benchmarks confirm that the decomposition explains where pairwise correction helps and where classwise correction alone suffices. These span text classification, visual recognition, and multimodal rare-disease diagnosis.

Zhanliang Wang, Hongzhuo Chen, Quan Minh Nguyen +2

University of Pennsylvania, Philadelphia, PA, USA · Children’s Hospital of Philadelphia, Philadelphia, PA, USA

Long-Tailed ClassificationLearning to RankResidual LearningPairwise Ranking
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Mar 30, 2026

Universal Approximation Constraints of Narrow ResNets: The Tunnel Effect

We analyze the universal approximation constraints of narrow Residual Neural Networks (ResNets) both theoretically and numerically. For deep neural networks without input space augmentation, a central constraint is the inability to represent critical points of the input-output map. We prove that this has global consequences for target function approximations and show that the manifestation of this defect is typically a shift of the critical point to infinity, which we call the ``tunnel effect'' in the context of classification tasks. While ResNets offer greater expressivity than standard multilayer perceptrons (MLPs), their capability strongly depends on the signal ratio between the skip and residual channels. We establish quantitative approximation bounds for both the residual-dominant (close to MLP) and skip-dominant (close to neural ODE) regimes. These estimates depend explicitly on the channel ratio and uniform network weight bounds. Low-dimensional examples further provide a detailed analysis of the different ResNet regimes and how architecture-target incompatibility influences the approximation error.

Christian Kuehn, Sara-Viola Kuntz, Tobias Wöhrer

Technical University of Munich, School of Computation, Information and Technology, Department of Mathematics, Boltzmannstraße 3, 85748 Garching, Germany · Munich Data Science Institute (MDSI), Garching, Germany · Munich Center for Machine Learning (MCML), München, Germany +1

Universal ApproximationNeural Network Approximation TheoryFunction ApproximationNeural Network ExpressivityResidual Learning
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Mar 23, 2026

GradAttn: Replacing Fixed Residual Connections with Task-Modulated Attention Pathways

Deep ConvNets suffer from gradient signal degradation as network depth increases, limiting effective feature learning in complex architectures. ResNet addressed this through residual connections, but these fixed short circuits cannot adapt to varying input complexity or selectively emphasize task-relevant features across network hierarchies. This study introduces GradAttn, a variation of the residual approach in CNNs that replaces the fixed residual connections with attention-controlled gradient flow. By extracting multi-scale CNN features at different depths and regulating them through self-attention, GradAttn dynamically weights shallow texture features and deep semantic representations. For representational analysis, we evaluated three GradAttn variants across eight diverse datasets: from natural images and medical imaging to fashion recognition. The results demonstrate that GradAttn outperforms ResNet-18 on five of eight datasets, achieving up to +11.07% accuracy improvement on FashionMNIST while maintaining a comparable network size. Gradient flow analysis reveals that controlled instabilities, introduced by attention, often coincide with improved generalization, challenging the assumption that perfect stability is optimal. Furthermore, positional encoding's effectiveness turned out to be dataset-dependent, with CNN hierarchies frequently encoding sufficient spatial structure. These findings render attention mechanisms as enablers of learnable gradient control, offering a new way for adaptive representation learning in deep neural architectures.

Soudeep Ghoshal, Himanshu Buckchash

Kalinga Institute of Industrial Technology, Bhubaneswar, India · IMC University of Applied Sciences Krems, Krems, Austria

Convolutional Neural NetworksAttention Residual ConnectionsResidual LearningMulti-Scale Attention
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Mar 21, 2026

Spectral-Sphere-Constrained Hyper-Connections

Hyper-Connections (HC) extend residual connections into multiple streams, employing residual matrices for cross-stream mixing to enrich model expressivity. However, unconstrained mixing disrupts the identity mapping property intrinsic to the residual connection, causing unstable training. To address this, Manifold-Constrained Hyper-Connections (mHC) and its variants restrict these matrices to be doubly stochastic via Sinkhorn-Knopp (SK) algorithm or permutation-based parameterizations. We reveal three limitations of this doubly stochastic constraint: (1) identity degeneration, where learned matrices collapse around the identity initialization and diminish cross-stream interactions, (2) a expressivity bottleneck, where the doubly stochastic constraint restricts the freedom of the subdominant spectrum of the residual matrices, preventing the model from selectively preserving or attenuating cross-stream variations, and (3) parameterization inefficiencies, manifesting as unstable SK iterations or the factorial-scaling overhead of permutation-based parameterizations. To overcome these flaws, we propose Spectral-Sphere-Constrained Hyper-Connections (s2\mathrm{s}^{2}HC). By confining residual matrices to a spectral norm sphere, s2\mathrm{s}^{2}HC restores free control over the subdominant spectrum, enabling the model to selectively preserve or attenuate cross-stream variations. This shift eliminates unstable SK iterations and factorial parameterization, enabling expressive, non-degenerate residual matrices while preserving training stability.

Zhaoyi Liu, Haichuan Zhang, Ang Li

University of Maryland, College Park · University of Utah

Hyper-ConnectionsResidual LearningConstrained Optimization
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Mar 16, 2026

Deep learning and the rate of approximation by flows

We investigate the dependence of the approximation capacity of deep residual networks on its depth in a continuous dynamical systems setting. This can be formulated as the general problem of quantifying the minimal time-horizon required to approximate a diffeomorphism by flows driven by a given family F\mathcal F of vector fields. We show that this minimal time can be identified as a geodesic distance on a sub-Finsler manifold of diffeomorphisms, where the local geometry is characterised by a variational principle involving F\mathcal F. This connects the learning efficiency of target relationships to their compatibility with the learning architectural choice. Further, the results suggest that the key approximation mechanism in deep learning, namely the approximation of functions by composition or dynamics, differs in a fundamental way from linear approximation theory, where linear spaces and norm-based rate estimates are replaced by manifolds and geodesic distances.

Jingpu Cheng, Qianxiao Li, Ting Lin +1

Department of Mathematics, National University of Singapore, 117543, Singapore · Department of Mathematics and Institute for Functional Intelligent Materials, National University of Singapore, 117543, Singapore · School of Mathematical Sciences, Peking University, 100871, China

Neural Network Approximation TheoryFunction ApproximationResidual Learning
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Mar 16, 2026

Attention Residuals

Residual connections with PreNorm are standard in modern LLMs, yet they accumulate all layer outputs with fixed unit weights. This uniform aggregation causes uncontrolled hidden-state growth with depth, progressively diluting each layer's contribution. We propose Attention Residuals (AttnRes), which replaces this fixed accumulation with softmax attention over preceding layer outputs, allowing each layer to selectively aggregate earlier representations with learned, input-dependent weights. To address the memory and communication overhead of attending over all preceding layer outputs for large-scale model training, we introduce Block AttnRes, which partitions layers into blocks and attends over block-level representations, reducing the memory footprint while preserving most of the gains of full AttnRes. Combined with cache-based pipeline communication and a two-phase computation strategy, Block AttnRes becomes a practical drop-in replacement for standard residual connections with minimal overhead. Scaling law experiments confirm that the improvement is consistent across model sizes, and ablations validate the benefit of content-dependent depth-wise selection. We further integrate AttnRes into the Kimi Linear architecture (48B total / 3B activated parameters) and pre-train on 1.4T tokens, where AttnRes mitigates PreNorm dilution, yielding more uniform output magnitudes and gradient distribution across depth, and improves downstream performance across all evaluated tasks.

Kimi Team, Guangyu Chen, Yu Zhang +34

Attention Residual ConnectionsResidual LearningLanguage Model Pretraining
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Mar 16, 2026

Ablate and Rescue: A Causal Analysis of Residual Stream Hyper-Connections

Multi-stream transformer architectures have recently been proposed as a promising direction for managing representation collapse and the vanishing gradient problem for residual connections, yet their internal mechanisms remain unexplored. In particular, the recently introduced Manifold-Constrained Hyper-Connections (mHC) architecture posits multiple residual streams with constrained interaction, but lacks in-depth mechanistic analysis. We present the first open-source mHC language model (https://huggingface.co/wgpeng/mhc-780m) and analyze the multiple-stream architecture with a suite of representation-level metrics and causal interventions to probe how parallel streams encode and utilize information. Specifically, we introduce a systematic stream ablation-and-rescue framework that enables direct causal comparison of residual streams during inference. Through targeted pairwise interventions and controlled recovery experiments, we distinguish functional redundancy from asymmetric utilization and reveal how information is distributed across streams beyond what is observable from representational similarity alone.

William Peng, Josheev Rai, Kevin Tseng +2

Stanford University · Georgia Institute of Technology · University of California, Berkeley +2

TransformerHyper-ConnectionsResidual LearningTransformer InterpretabilityCausal Interventions in Language Models
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Mar 12, 2026

Learn Structure, Adapt on the Fly: Multi-Scale Residual Learning and Online Adaptation for Aerial Manipulators

Autonomous Aerial Manipulators (AAMs) are inherently coupled, nonlinear systems that exhibit nonstationary and multiscale residual dynamics, particularly during manipulator reconfiguration and abrupt payload variations. Conventional analytical dynamic models rely on fixed parametric structures, while static data-driven model assume stationary dynamics and degrade under configuration changes and payload variations. Moreover, existing learning architectures do not explicitly factorize cross-variable coupling and multi-scale temporal effects, conflating instantaneous inertial dynamics with long-horizon regime evolution. We propose a predictive-adaptive framework for real-time residual modeling and compensation in AAMs. The core of this framework is the Factorized Dynamics Transformer (FDT), which treats physical variables as independent tokens. This design enables explicit cross-variable attention while structurally separating short-horizon inertial dependencies from long-horizon aerodynamic effects. To address deployment-time distribution shifts, a Latent Residual Adapter (LRA) performs rapid linear adaptation in the latent space via Recursive Least Squares, preserving the offline nonlinear representation without prohibitive computational overhead. The adapted residual forecast is directly integrated into a residual-compensated adaptive controller. Real-world experiments on an aerial manipulator subjected to unseen payloads demonstrate higher prediction fidelity, accelerated disturbance attenuation, and superior closed-loop tracking precision compared to state-of-the-art learning baselines, all while maintaining strict real-time feasibility.

Samaksh Ujjawal, Naveen Sudheer Nair, Shivansh Pratap Singh +3

1Robotics Research Center, IIIT Hyderabad, India. · Department of Computer Science, University of Manchester, UK. · 3Newcastle University, UK.

Aerial RoboticsTest-Time AdaptationLatent Dynamics ModelingResidual LearningAdaptive ControlMultiscale Time-Series Modeling
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Mar 11, 2026

SCORE: Replacing Layer Stacking with Contractive Recurrent Depth

Residual connections are central to modern deep neural networks, enabling stable optimization and efficient information flow across depth. In this work, we propose SCORE (Skip-Connection ODE Recurrent Embedding), a discrete recurrent alternative to classical layer stacking. Instead of composing multiple independent layers, SCORE iteratively applies a single shared neural block using an ODE (Ordinary Differential Equation)-inspired contractive update: ht+1 = (1 - dt) * ht + dt * F(ht) This formulation can be interpreted as a depth-by-iteration refinement process, where the step size dt explicitly controls stability and update magnitude. Unlike continuous Neural ODE approaches, SCORE uses a fixed number of discrete iterations and standard backpropagation without requiring ODE solvers or adjoint methods. We evaluate SCORE across graph neural networks (ESOL molecular solubility), multilayer perceptrons, and Transformer-based language models (nanoGPT). Across architectures, SCORE generally improves convergence speed and often accelerates training. SCORE is reducing parameter count through shared weights. In practice, simple Euler integration provides the best trade-off between computational cost and performance, while higher-order integrators yield marginal gains at increased compute. These results suggest that controlled recurrent depth with contractive residual updates offers a lightweight and effective alternative to classical stacking in deep neural networks.

Guillaume Godin

1Osmo Labs PBC New York, USA

Residual LearningRecurrent Neural NetworksNeural Network Optimization
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Mar 10, 2026

Mitigating Frequency Learning Bias in Quantum Models via Multi-Stage Residual Learning

Quantum machine learning models based on parameterized circuits can be viewed as Fourier series approximators. However, they often struggle to learn functions with multiple frequency components, particularly high-frequency or non-dominant ones; a phenomenon we term the quantum Fourier parameterization bias. Inspired by recent advances in classical Fourier neural operators (FNOs), we adapt the multi-stage residual learning idea to the quantum domain, iteratively training additional quantum modules on the residuals of previous stages. We evaluate our method on a synthetic benchmark composed of spatially localized frequency components with diverse envelope shapes (Gaussian, Lorentzian, triangular). Systematic experiments show that the number of qubits, the encoding scheme, and residual learning are all crucial for resolving multiple frequencies; residual learning alone can improve test MSE significantly over a single-stage baseline trained for the same total number of epochs. Our work provides a practical framework for enhancing the spectral expressivity of quantum models and offers new insights into their frequency-learning behavior.

Ammar Daskin

Department of Computer Engineering, Istanbul Medeniyet University, Istanbul, Turkiye, 34000

Quantum Machine LearningQuantum Neural NetworksResidual Learning
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Mar 4, 2026

Observationally Informed Adaptive Causal Experimental Design

Randomized Controlled Trials (RCTs) represent the gold standard for causal inference yet remain a scarce resource. While large-scale observational data is often available, it is utilized only for retrospective fusion, and remains discarded in prospective trial design due to bias concerns. We argue this "tabula rasa" data acquisition strategy is fundamentally inefficient. In this work, we propose Active Residual Learning, a new paradigm that leverages the observational model as a foundational prior. This approach shifts the experimental focus from learning target causal quantities from scratch to efficiently estimating the residuals required to correct observational bias. To operationalize this, we introduce the R-Design framework. Theoretically, we establish two key advantages: (1) a structural efficiency gap, proving that estimating smooth residual contrasts admits strictly faster convergence rates than reconstructing full outcomes; and (2) information efficiency, where we quantify the redundancy in standard parameter-based acquisition (e.g., BALD), demonstrating that such baselines waste budget on task-irrelevant nuisance uncertainty. We propose R-EPIG (Residual Expected Predictive Information Gain), a unified criterion that directly targets the causal estimand, minimizing residual uncertainty for estimation or clarifying decision boundaries for policy. Experiments on synthetic and semi-synthetic benchmarks demonstrate that R-Design significantly outperforms baselines, confirming that repairing a biased model is far more efficient than learning one from scratch.

Erdun Gao, Liang Zhang, Jake Fawkes +5

Australian Institute for Machine Learning, Adelaide University · Hong Kong University of Science and Technology (Guangzhou) · University College London +3

Active Experimental DesignCausal InferenceCausal Effect EstimationAdaptive SamplingBayesian Experimental DesignResidual Learning
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Feb 28, 2026

Bi-cLSTM: Residual-Corrected Bidirectional LSTM for Aero-Engine RUL Estimation

Accurate Remaining Useful Life (RUL) prediction is a key requirement for effective Prognostics and Health Management (PHM) in safety-critical systems such as aero-engines. Existing deep learning approaches, particularly LSTM-based models, often struggle to generalize across varying operating conditions and are sensitive to noise in multivariate sensor data. To address these challenges, we propose a novel Bidirectional Residual Corrected LSTM (Bi-cLSTM) model for robust RUL estimation. The proposed architecture combines bidirectional temporal modeling with an adaptive residual correction mechanism to iteratively refine sequence representations. In addition, we introduce a condition-aware preprocessing pipeline incorporating regime-based normalization, feature selection, and exponential smoothing to improve robustness under complex operating environments. Extensive experiments on all four subsets of the NASA C-MAPSS dataset demonstrate that the proposed Bi-cLSTM consistently outperforms LSTM-based baselines and achieves competitive state-of-the-art performance, particularly in challenging multi-condition scenarios. These results highlight the effectiveness of combining bidirectional temporal learning with residual correction for reliable RUL prediction.

Rafi Hassan Chowdhury, Nabil Daiyan, Faria Ahmed +2

Department of Computer Science and Engineering, Rajshahi University of Engineering and Technology, Rajshahi, Bangladesh · Department of Aeronautical Engineering, Military Institute of Science and Technology, Dhaka 1216, Bangladesh

RUL EstimationPredictive MaintenanceRecurrent Neural NetworksResidual Learning
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Feb 24, 2026

On the Generalization Behavior of Deep Residual Networks From a Dynamical System Perspective

Deep neural networks (DNNs) have significantly advanced machine learning, with model depth playing a central role in their successes. The dynamical system modeling approach has recently emerged as a powerful framework, offering new mathematical insights into the structure and learning behavior of DNNs. In this work, we establish generalization error bounds for both discrete- and continuous-time residual networks (ResNets) by combining Rademacher complexity, flow maps of dynamical systems, and the convergence behavior of ResNets in the deep-layer limit. The resulting bounds are of order O(1/S)O(1/\sqrt{S}) with respect to the number of training samples SS, and include a structure-dependent negative term, yielding depth-uniform and asymptotic generalization bounds under milder assumptions. These findings provide a unified understanding of generalization across both discrete- and continuous-time ResNets, helping to close the gap in both the order of sample complexity and assumptions between the discrete- and continuous-time settings.

Jinshu Huang, Mingfei Sun, Chunlin Wu

School of Mathematical Sciences, Nankai University, Tianjin, China

Neural Network GeneralizationGeneralization BoundsResidual LearningStatistical Learning Theory
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Feb 20, 2026

Scientific Knowledge-Guided Machine Learning for Vessel Power Prediction: A Comparative Study

Accurate prediction of main engine power is essential for vessel performance optimization, fuel efficiency, and compliance with emission regulations. Conventional machine learning approaches, such as Support Vector Machines, variants of Artificial Neural Networks (ANNs), and tree-based methods like Random Forests, Extra Tree Regressors, and XGBoost, can capture nonlinearities but often struggle to respect the fundamental propeller law relationship between power and speed, resulting in poor extrapolation outside the training envelope. This study introduces a hybrid modeling framework that integrates physics-based knowledge from sea trials with data-driven residual learning. The baseline component, derived from calm-water power curves of the form P=cVnP = cV^n, captures the dominant power-speed dependence, while another, nonlinear, regressor is then trained to predict the residual power, representing deviations caused by environmental and operational conditions. By constraining the machine learning task to residual corrections, the hybrid model simplifies learning, improves generalization, and ensures consistency with the underlying physics. In this study, an XGBoost, a simple Neural Network, and a Physics-Informed Neural Network (PINN) coupled with the baseline component were compared to identical models without the baseline component. Validation on in-service data demonstrates that the hybrid model consistently outperformed a pure data-driven baseline in sparse data regions while maintaining similar performance in populated ones. The proposed framework provides a practical and computationally efficient tool for vessel performance monitoring, with applications in weather routing, trim optimization, and energy efficiency planning.

Orfeas Bourchas, George Papalambrou

Laboratory of Marine Engineering N.T.U.A.

Physics-Informed MLResidual Learning
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Feb 20, 2026

JPmHC Dynamical Isometry via Orthogonal Hyper-Connections

Recent advances in deep learning, exemplified by Hyper-Connections (HC), have expanded the residual connection paradigm by introducing wider residual streams and diverse connectivity patterns. While these innovations yield significant performance gains, they compromise the identity mapping property of residual connections, leading to training instability, limited scalability, and increased memory overhead. To address these challenges, we propose JPmHC (Jacobian-spectrum Preserving manifold-constrained Hyper-Connections), a framework that replaces identity skips with a trainable linear mixer acting on n parallel streams while explicitly controlling gradient conditioning. By constraining the mixer M on operator-norm-bounded manifolds (e.g., bistochastic, Stiefel, Grassmann), JPmHC prevents gradient pathologies and enhances stability. JPmHC introduces three key contributions: (i) a free-probability analysis that predicts Jacobian spectra for structured skips, providing actionable design rules for mixer selection; (ii) memory-efficient implicit differentiation for fixed-point projections, reducing activation memory and synchronization overhead; and (iii) a Stiefel-constrained mixer via Cayley transforms, ensuring orthogonality without post-hoc normalization. Empirical evaluations on ARC-AGI demonstrate that JPmHC achieves faster convergence, higher accuracy, and lower computational cost compared to bistochastic baselines, with a rank-pp Grassmannian variant tracking between the two -- consistent with the spectral theory predictions. As a flexible and scalable extension of HC, JPmHC advances spectrum-aware, stable, and efficient deep learning, offering insights into topological architecture design and foundational model evolution. \newline \newline

Biswa Sengupta, Jinhua Wang, Leo Brunswic

LLM Suite Team, JP Morgan Chase & Co.

Hyper-ConnectionsResidual Learning
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Feb 20, 2026

Turbo Connection: Reasoning as Information Flow from Higher to Lower Layers

Complex problems, whether in math, logic, or planning, are solved by humans through a sequence of steps where the result of one step informs the next. In this work, we adopt the perspective that the reasoning power of Transformers is fundamentally limited by a fixed maximum number of steps along any latent path of computation. To address this, we introduce Turbo Connection (TurboConn), a novel architecture that overcomes the fixed-depth constraint by routing multiple residual connections from the higher-layer hidden states of each token tt to the lower layers of token t+1t+1. Fine-tuning pre-trained LLMs with our method not only yields accuracy gains of 0.9% to over 10% on benchmarks like GSM8K, Parity, and multi-step arithmetic, but also demonstrates that the density of these backward connections is critical; our dense interaction significantly outperforms "sparse" alternatives that only pass a single hidden state or vector. Notably, TurboConn can be integrated into pre-trained LLMs to overcome task-specific plateaus: while a fine-tuned Qwen-3-1.7B achieves only 53.78% on Parity, adding our architectural modification enables the model to reach 100% accuracy, all without the necessity to retrain the full model from scratch or sophisticated curriculum learning. Our results provide strong empirical evidence that the depth of the computational path is a key factor in reasoning ability, also offering a new mechanism to enhance LLMs without significantly affecting generation latency.

Mohan Tang, Sidi Lu

UCLA.

TransformerLarge Reasoning ModelsNumerical Reasoning in Language ModelsResidual Learning
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Feb 18, 2026

HPMixer: Hierarchical Patching for Multivariate Time Series Forecasting

In long-term multivariate time series forecasting, effectively capturing both periodic patterns and residual dynamics is essential. To address this within standard deep learning benchmark settings, we propose the Hierarchical Patching Mixer (HPMixer), which models periodicity and residuals in a decoupled yet complementary manner. The periodic component utilizes a learnable cycle module [7] enhanced with a nonlinear channel-wise MLP for greater expressiveness. The residual component is processed through a Learnable Stationary Wavelet Transform (LSWT) to extract stable, shift-invariant frequency-domain representations. Subsequently, a channel-mixing encoder models explicit inter-channel dependencies, while a two-level non-overlapping hierarchical patching mechanism captures coarse- and fine-scale residual variations. By integrating decoupled periodicity modeling with structured, multi-scale residual learning, HPMixer provides an effective framework. Extensive experiments on standard multivariate benchmarks demonstrate that HPMixer achieves competitive or state-of-the-art performance compared to recent baselines.

Jung Min Choi, Vijaya Krishna Yalavarthi, Lars Schmidt-Thieme

ISMLL, University of Hildesheim

Multivariate Time Series ForecastingLong-Term Time Series ForecastingMultiscale Time-Series ModelingResidual Learning
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Feb 16, 2026

Residual Connections and the Causal Shift: Uncovering a Structural Misalignment in Transformers

Large Language Models (LLMs) are trained with next-token prediction, implemented in autoregressive Transformers via causal masking for parallelism. This creates a subtle misalignment: residual connections tie activations to the current token, while supervision targets the next token, potentially propagating mismatched information if the current token is not the most informative for prediction. In this work, we empirically localize this input-output alignment shift in pretrained LLMs, using decoding trajectories over tied embedding spaces and similarity-based metrics. Our experiments reveal that the hidden token representations switch from input alignment to output alignment deep within the network. Motivated by this observation, we propose a lightweight residual-path mitigation based on residual attenuation, implemented either as a fixed-layer intervention or as a learnable gating mechanism. Experiments on multiple benchmarks show that these strategies alleviate the representation misalignment and yield improvements, providing an efficient and general architectural enhancement for autoregressive Transformers.

Jonathan Lys, Vincent Gripon, Bastien Pasdeloup +4

IMT Atlantique, Lab-STICC, UMR CNRS 6285, Brest, France · Sony Europe Ltd. Stuttgart Technology Center, EUREC, Germany

TransformerResidual LearningDecoder-Only Language Models
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