arXiv subjects

Gitta Kutyniok

Publications and source records attributed to Gitta Kutyniok.

At least 19 recordsLinked to original sources

arXiv2026

Formalizing the Intrinsic Dimension of Objective Landscapes by a Generalized Notion of Co-dimension

In this work, we formalize the concept of intrinsic dimension of objective landscapes introduced in Li et al. (2018), which measures the structural complexity of learning problems by training along randomly generated subspaces. We define the intrinsic dimension of a set $Θ\subset\mathbb{R}^p$ as the minimal dimension $d_{\mathrm{int}}(Θ)$ such that a random plane of dimension $d_{\mathrm{int}}(Θ)$ in $\mathbb{R}^p$ intersects $Θ$ with probability one. We show that it is a generalized notion of co-dimension that depends on the global geometry of $Θ$. We study mathematical properties of the intrinsic dimension and develop techniques to estimate the intrinsic dimension of sets $Θ$ that arise in learning problems. More specifically, we derive upper and/or lower bounds where $Θ$ is the set of optimal parameters in student-teacher settings with shallow and deep ReLU networks, including finite and infinite sample cases, and with and without regularization. We discuss how our theory leads to novel types of generalization bounds, thereby showing that the intrinsic dimension is a measure of learning complexity. Our results connect techniques from seemingly different areas of machine learning research such as classification of symmetries of ReLU networks and stochastic geometry.

stat.ML↗

arXiv2026

When is a System Discoverable from Data? Discovery Requires Chaos

The deep learning revolution has enabled major AI advances in physical sciences, particularly in discovering dynamical systems from observational data. Yet the reliability of learned surrogates and symbolically discovered models is often undermined by non-uniqueness: models may fit observed trajectories perfectly, but lack genuine predictive power. We show that chaos, despite its association with unpredictability, is often what makes governing equations discoverable. We prove that the classical Lorenz system is analytically discoverable and verify experimentally that for many systems out-of-domain generalization emerges precisely at chaos onset. Conservative systems, by contrast, are inherently non-discoverable from data alone. The prevalence of chaotic systems in benchmarks may have inadvertently obscured this limitation. These findings help explain why AI excels at weather forecasting, yet struggles in many engineering applications: the systems most amenable to discovery (chaotic ones) are often least amenable to control.

math.DS↗

arXiv2026

Copy First, Translate Later: Interpreting Translation Dynamics in Multilingual Pretraining

Large language models exhibit impressive cross-lingual capabilities. However, prior work analyzes this phenomenon through isolated factors and at sparse points during training, limiting our understanding of how cross-lingual generalization emerges--particularly in the early phases of learning. To study the early trajectory of linguistic and translation capabilities, we pretrain a multilingual 1.7B model on nine diverse languages, capturing checkpoints at a much finer granularity. We use word-level translation as a testbed, introducing a novel dataset to trace how translation develops over training through behavioral analyses, model-component analysis, and parameter-based ablations. We find that the model quickly acquires basic linguistic capabilities in parallel with token-level copying, while translation develops in two distinct phases: an initial phase dominated by copying and surface-level similarities, and a second phase in which more generalizing translation mechanisms are developed while copying is refined. Together, these findings provide a fine-grained view of how cross-lingual generalization develops during multilingual pretraining.

cs.CL↗

arXiv2026

SCORE: Spectral Correlation Estimation for Multivariate Gaussians

Neural network-based predictive modeling with high-dimensional structured Gaussian targets requires an efficient and numerically stable, yet expressive approximation of the covariance matrix. We propose SCORE: a scalable framework, combining scoring rule training with an expressive covariance approximation learned in spectral space. For $d$-dimensional data, the learning task is decomposed into learning the marginal distributions and learning a structured correlation matrix, which enables dense dependencies with linear storage and $\mathcal{O}(d\log d)$ cost. We utilize the closed form Gaussian kernel score for training, which remains defined even for degenerate covariances and admits bounded gradients during optimization. We characterize kernel scores under invertible transforms and prove exact invariance under unitary transforms. At population level, our two-level objective recovers the true marginals and projects the target correlation onto the representable class; finite-sample PAC bounds show that the errors of the two stages enter additively. We evaluate our model on a variety of tasks with a commonly assumed Gaussian domain: Time-series forecasting, monocular depth estimation, and spatial weather prediction, showing improved performance at lower computational cost.

cs.LG↗

arXiv2026

Understanding Multimodality in Generative Behavioral Cloning

Behavioral cloning becomes challenging when the same observation admits several valid actions. We study how generative behavioral-cloning policies represent such multimodal expert behavior and identify different bottlenecks across model parameterizations. For latent-variable policies, preserving demonstrated modes requires action-conditioned information in the latent representation. Excessive posterior-prior regularization can suppress this information and prevent the policy from distinguishing demonstrated modes. Weaker or aggregate regularization can preserve mode information, but shifts the challenge to ensuring that the deployment-time prior covers the relevant latent regions. For action-space generative policies, multimodality is constrained by the smoothness of the base-to-action transport: a map with a small Lipschitz constant cannot assign substantial probability to many well-separated modes. Covering many modes therefore requires either sharp transitions in base space or off-support bridge regions in action space. Experiments on synthetic multimodal navigation and a physical-robot bimodal manipulation task support these mechanisms. In contrast, our analysis reveals limited conditional multimodality in standard robotic simulation benchmarks, where deterministic regression remains competitive.

cs.LG↗

arXiv2026

Scale Sensitivity in Low-Bit Post-Training Quantization: Curvature of the Quantization Error Landscape

Post-training quantization (PTQ) methods in the GPTQ family minimize a layer-wise reconstruction error on a uniform grid whose scale must be chosen; the common max-based choice degrades sharply at low bit-widths. We study how sensitive this objective is to the scale. For a layer with i.i.d. Gaussian weights and calibration activations of sufficiently large effective rank, we prove that, as the width grows, the normalized round-to-nearest loss converges with high probability, uniformly over all scales, to the mean-squared error of a uniform quantizer applied to a standard Gaussian; we verify the effective-rank condition for wide, randomly initialized MLPs with odd Lipschitz activations and isotropic Gaussian calibration data. The limiting objective has a unique nondegenerate minimizer, whose scale decreases strictly with the number of levels and whose curvature with respect to relative scale errors decays approximately exponentially with the bit-width. GPTQ experiments on five LLMs show the same trend: the scale rule changes perplexity substantially at 2--3 bits and negligibly from 6 bits on, and a local measure of GPTQ scale sensitivity decreases with bit-width in line with the Gaussian curvature. The Gaussian-optimal scale fails on raw weights; after Hadamard incoherence processing it matches the best searched rule at 3 bits and above without any search, but remains clearly worse at 2 bits.

cs.LG↗

arXiv2026

TACTIC: Understanding Tactile Encoders and Conditioning for Contact-rich Robot Manipulation Policies

Tactile information is essential for contact-rich manipulation tasks in robotics. Vision-based tactile sensors make it particularly easy to design end-to-end manipulation policies with tactile sensing, as they enable the use of existing encoders from computer vision. However, this has led to a huge variety of architectures, training datasets, and evaluation protocols, making it difficult to determine which design choices best encode touch. In this work, we address this gap and present a comprehensive study of tactile encoders and fusion strategies across various contact-rich manipulation tasks in real-world experiments. To enable a controlled comparison, we train and evaluate all models under the same pipeline and experimental setup, comprising more than 2000 real-world rollouts. Our results go beyond other studies that only compare simulation performance, which does not necessarily translate to real-world settings, where large-scale evaluations are needed to obtain reliable statistics. Our key finding is that there is no universally optimal representation or fusion strategy for encoding visual-tactile. Instead, the best encoder backbone and fusion scheme depend strongly on the task.

cs.RO↗

arXiv2026

Polyhedral Geometry of Time-to-First-Spike Neural Networks

We study the expressivity of spiking neural networks, which provide a natural framework for asynchronous, event-driven computation complementary to conventional feedforward neural networks. We consider the time-to-first-spike model in a setting for which the input-output map is continuous and piecewise linear, with affine pieces governed by causal feasibility constraints that determine which presynaptic spikes occur before a neuron fires. We first show that each neuron's firing time admits a maxout-like representation with exponentially many, highly constrained affine pieces. We then formalize causal regions as polyhedral regions with fixed causal sets and derive upper and lower bounds on the maximal number of causal regions in both shallow and multilayer feedforward spiking networks. Our theoretical and experimental results show that spiking networks can generate richer partitions of the input space than conventional feedforward ReLU networks.

cs.LG↗

arXiv2026

Functional Degeneracy in Neural Networks: Measurement and Pruning

A central question in modern machine learning is how much a trained model can be compressed without changing its behavior, to reduce the memory, compute and energy required to deploy it. To study this, we quantify functional degeneracy through the behavioral recovery rank, defined as the number of leading behavioral-Hessian eigendirections required to recover a trained model's performance. Using the behavioral recovery rank as a geometric benchmark for compression, we find that structural and magnitude pruning retain more degrees of freedom, even after the task is saturated. This gap suggests that functional redundancy is distributed across parameter directions and is not exposed by individual weights or neurons.

cs.LG↗

arXiv2026

Towards an Expressivity-Normalized Energy-Demand Comparison of ANNs and SNNs

Spiking neural networks (SNNs) are often regarded as energy-efficient alternatives to artificial neural networks (ANNs), yet their advantage depends critically on both network architecture and data properties. We develop an analytical framework to compare fully-connected ReLU ANNs and integrate-and-fire SNNs for time-series data with respect to their theoretical energy efficiency at matched expressive capacity. By relating an inference-energy model to theoretical bounds on representational expressivity, we derive an expressivity-normalized efficiency ratio and explicit thresholds in network width, spike sparsity, and ANN depth scaling. Our analysis characterizes the regimes in which event-driven computation offsets the temporal overhead of SNNs, providing capacity-aware principles for designing energy-efficient temporal networks. It shows that ANNs exceed SNNs in expressivity-normalized efficiency only in specific regimes.

cs.LG↗

arXiv2026

Graph Representational Learning: When Does More Expressivity Hurt Generalization?

Graph Neural Networks (GNNs) are powerful tools for learning on structured data, yet the relationship between their expressivity and predictive performance remains unclear. We introduce a family of premetrics that capture different degrees of structural similarity between graphs and relate these similarities to generalization, and consequently, the performance of expressive GNNs. By considering a setting where graph labels are correlated with structural features, we derive generalization bounds that depend on the distance between training and test graphs, model complexity, and training set size. These bounds reveal that more expressive GNNs may generalize worse unless their increased complexity is balanced by a sufficiently large training set or reduced distance between training and test graphs. Our findings relate expressivity and generalization, offering theoretical insights supported by empirical results.

cs.LG↗

arXiv2026

Symbolic Recovery of Differential Equations: The Identifiability Problem

Symbolic recovery of differential equations is the ambitious attempt at automating the derivation of governing equations with the use of machine learning techniques. In contrast to classical methods which assume the structure of the equation to be known and focus on the estimation of specific parameters, these algorithms aim to learn the structure and the parameters simultaneously. While the uniqueness and, therefore, the identifiability of parameters of governing equations are a well-addressed problem in the field of parameter estimation, it has not been investigated for symbolic recovery. However, this problem should be even more present in this field since the algorithms aim to cover larger spaces of governing equations. In this paper, we investigate under which conditions a solution of a differential equation does not uniquely determine the equation itself. For various classes of differential equations, we provide both necessary and sufficient conditions for a function to uniquely determine the corresponding differential equation. We then use our results to devise numerical algorithms aiming to determine whether a function solves a differential equation uniquely. Finally, we provide extensive numerical experiments showing that our algorithms can indeed guarantee the uniqueness of the learned governing differential equation, without assuming any knowledge about the analytic form of function, thereby ensuring the reliability of the learned equation.

cs.LG↗

arXiv2026

An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression

Uncertainty quantification is crucial in machine learning, yet most (axiomatic) studies of uncertainty measures focus on classification, leaving a gap in regression settings with limited formal justification and evaluations. In this work, we provide a formal way of representing uncertainty in continuous space, using a general parametric formulation, allowing for tractable analysis and evaluation of uncertainty measures. Within this framework, we propose a set of axioms that enable rigorous assessment of total, aleatoric, and epistemic uncertainty measures. Together, this allows for a theoretical examination of uncertainty measures and their corresponding properties. As a specific example, we compare the widely used entropy- and variance-based measures with respect to established predictive models and analyze their limitations and challenges in uncertainty quantification. Our work provides a principled way to understand and develop uncertainty measures in supervised regression, offering theoretical insights and practical guidelines for reliable uncertainty assessment.

cs.LG↗

arXiv2026

Uncertainty Quantification for Regression: A Unified Framework based on kernel scores

Regression tasks, notably in safety-critical domains, require reliable uncertainty quantification, yet the literature remains largely classification-focused. To address this, we introduce a family of measures for total, aleatoric, and epistemic uncertainty in multivariate regression based on strictly proper kernel scores. The framework provides a principled recipe for designing new uncertainty measures whose behavior, such as tail sensitivity or out-of-distribution responsiveness, is governed by the choice of the underlying kernel, while also encompassing existing measures under a joint analysis. We prove explicit correspondences between properties of the kernel and behavior of resulting uncertainty measures, yielding concrete design guidelines for practitioners. Extensive experiments across structured regression tasks, including spatial and functional domains, demonstrate effectiveness on downstream tasks such as out-of-distribution detection and active learning, and reveal that different kernel choices lead to distinct trade-offs, offering practitioners guidance for task-specific selection.

cs.LG↗

arXiv2026

A Variational Framework for the Complexity of PDE Solutions

Partial Differential Equations (PDEs) are fundamental mathematical models for describing physical phenomena, yet most PDEs of practical interest require numerical approximations. The feasibility of such methods is constrained by existing computational models. Since digital computers are the primary realizations of numerical computations, and Turing machines define their theoretical limits, computability of PDE solutions is of fundamental significance. It provides a rigorous framework to distinguish equations that are effectively solvable from those that encode undecidable or non-computable behavior. Once computability is established, complexity theory quantifies the resources required to approximate PDE solutions. In this work, we present a novel framework based on least-squares variational formulations and associated gradient flows to analyze the computability and complexity of PDE solutions from an optimization perspective. Our approach approximates PDE solution operators via discrete gradient flows, linking PDE properties, such as coercivity, ellipticity, and convexity, to solution complexity. Within this setting, we characterize representation- and discretization-dependent sufficient conditions for regimes where PDEs admit polynomial-time approximations, as well as regimes exhibiting complexity blowup, where polynomial-time input data produce solutions with super-polynomial complexity. In summary, this paper develops a variational framework for analyzing computability and computational complexity of PDE solution classes. The results show how PDE structure and solution regularity influence their complexity, by establishing sufficient conditions for computability and complexity bounds. Beyond the theoretical characterization, the framework provides guidelines for effective numerical methods and contributes to understanding the limitations of digital computation for PDE problems.

math.NA↗

arXiv2026

Conflicting Biases at the Edge of Stability: Norm versus Sharpness Regularization

The remarkable generalization properties of overparameterized networks are often attributed to implicit biases, such as norm minimization at small learning rates and low sharpness in the Edge-of-Stability regime. In this work, we argue that a comprehensive understanding of the generalization performance of gradient descent requires analyzing the interaction between these various forms of implicit regularization. We empirically demonstrate that the learning rate interpolates between low parameter norm and low sharpness of the trained model. We furthermore prove that neither implicit bias alone minimizes the generalization error for diagonal linear networks trained on a simple regression task. These findings demonstrate that focusing on a single implicit bias is insufficient to explain good generalization, and they motivate a broader view of implicit regularization that captures the dynamic trade-off between norm and sharpness induced by non-negligible learning rates.

cs.LG↗

arXiv2026

CHEM: Estimating and Understanding Hallucinations in Deep Learning for Image Processing

Deep learning-based methods have recently achieved significant success in image reconstruction problems. However, challenges have emerged, as these methods may generate unrealistic artifacts or hallucinations, which can interfere with analysis in safety-critical scenarios. This paper introduces a framework for quantifying and characterizing hallucinated artifacts in image reconstruction models. The proposed method, termed the Conformal Hallucination Estimation Metric (CHEM), enables the identification of hallucination-prone regions in model predictions. It leverages wavelet and shearlet representations to localize such regions at the level of image features, and uses conformalized quantile regression to assess hallucination levels in a distribution-free manner. A theoretical analysis is provided, characterizing the sensitivity of CHEM to hallucinated artifacts and its relationship to the mean squared error. Building on these insights and adopting a viewpoint grounded in approximation theory, we investigate why U-shaped networks, widely used architectures for image reconstruction, tend to hallucination-prone predictions. We assess the effectiveness of the proposed approach on astronomical image deconvolution using the CANDELS dataset with architectures such as U-Net, SwinUNet, and Learnlets, and on natural image super-resolution using the DIV2K dataset with models such as DRUNet, Unfolded DRS, RAM, and DPS.

cs.CV↗

arXiv2026

Sparse-Aware Neural Networks for Nonlinear Functionals: Mitigating the Exponential Dependence on Dimension

Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited interpretability. This work investigates how sparsity can help address these challenges in functional learning, a central ingredient in operator learning. We propose a framework that employs convolutional architectures to extract sparse features from a finite number of samples, together with deep fully connected networks to effectively approximate nonlinear functionals. Using universal discretization methods, we show that sparse approximators enable stable recovery from discrete samples. In addition, both the deterministic and the random sampling schemes are sufficient for our analysis. These findings lead to improved approximation rates and reduced sample sizes in various function spaces, including those with fast frequency decay and mixed smoothness. They also provide new theoretical insights into how sparsity can alleviate the curse of dimensionality in functional learning.

cs.LG↗