Many learning algorithms can be represented as Markov processes, and understanding their generalization error is a central topic in learning theory. For specific continuous-time noisy algorithms, a prominent analysis technique relies on information-theoretic tools and the so-called ``entropy flow'' method. This technique is compatible with a broad range of assumptions and leverages the convergence properties of learning dynamics to produce meaningful generalization bounds, which can also be informative or extend to discrete-time settings. Despite their success, existing entropy flow formulations are limited to specific noise and algorithm structures (\eg, Langevin dynamics). In this work, we exploit new technical tools to extend its applicability to all learning algorithms whose iterative dynamics is governed by a time-homogeneous Markov process. Our approach builds on a principled continuous-time approximation of Markov algorithms and introduces a new, exact entropy flow formula for such processes. Within this unified framework, we establish novel connections to a well-studied family of modified logarithmic Sobolev inequalities, which we use to connect the generalization error to the ergodic properties of Markov processes. Finally, we provide a detailed analysis of all the terms appearing in our theory and demonstrate its effectiveness by deriving new generalization bounds for several concrete algorithms.
Benjamin Dupuis, Maxime Haddouche, George Deligiannidis et al.· 0 citations
In this article, we introduce a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.
Christian Beck, Sebastian Becker, Patrick Cheridito et al.· 0 citations
Transformers have achieved remarkable successes across a wide range of applications, yet the theoretical foundation of their model efficiency remains underexplored. In this work, we investigate how the model parameters -- mainly attention heads and head dimensions -- should be allocated across layers to balance expressivity and efficiency. We first provide mathematical analysis on the role of early layers in information extraction from an approximation perspective, with a theoretical characterization on the trade-off between the number of heads and head dimension under a fixed parameter budget. In addition, we uncover and prove the \emph{saturation} behavior of softmax activations: Continuously increasing head dimensions can lead to diminishing returns in learning errors, particularly for long sequences. Supported by both theory and experiments, this saturation pattern suggests that later layers can operate more efficiently with reduced parameters. Combining these insights, we propose principled strategies for allocating attention heads and dimensions across Transformers' layers, shedding light on theoretically-grounded model efficiency of Transformer-based architectures.
Ruoxi Yu, Haotian Jiang, Jingpu Cheng et al.· 0 citations
Efficiently scaling Large Language Models (LLMs) necessitates exploring alternatives to dominant autoregressive (AR) methods, with Masked Diffusion Models (MDMs) emerging as candidates. However, comparing AR (typically decoder-only) and MDM (often encoder-only) paradigms is confounded by differing architectures, obscuring true algorithmic and efficiency trade-offs. This research decouples these factors by evaluating MDMs within a decoder-only framework to: (1) Equitably compare MDM (as Any-Order AR) and standard AR paradigms through discrepancies on orders. (2) Investigate MDM architectural impacts on computational efficiency. We show decoder-only MDMs, despite a larger modeling space, can achieve significant inference speedups ($\sim25\times$) and comparable perplexity with techniques like temperature annealing, offering a path to reduced inference compute. This work provides insights for developing more computationally efficient foundation models by disentangling core modeling choices from architectural influences. Code is available at https://github.com/scxue/AO-GPT-MDM.
Shuchen Xue, Tianyu Xie, Tianyang Hu et al.· 0 citations
Reach audiences
Advertise in front of researchers, engineers, and readers.
Causal discovery is a difficult problem that typically relies on strong assumptions on the data-generating model, such as non-Gaussianity. In practice, many modern applications provide multiple related views of the same system, which has rarely been considered for causal discovery. Here, we leverage this multi-view structure to achieve causal discovery with weak assumptions. We propose a multi-view linear Structural Equation Model (SEM) that extends the well-known framework of non-Gaussian disturbances by alternatively leveraging correlation over views. We prove the identifiability of the model for acyclic SEMs. Subsequently, we propose several multi-view causal discovery algorithms, inspired by single-view algorithms (DirectLiNGAM, PairwiseLiNGAM, and ICA-LiNGAM). The new methods are validated through simulations and applications on neuroimaging data, where they enable the estimation of causal graphs between brain regions.
Ambroise Heurtebise, Omar Chehab, Pierre Ablin et al.· 0 citations
The expansion of time-series data from sensors and monitoring systems has made compact representations increasingly important. Such representations should retain signal structure while cutting storage, transmission and computation costs. Adaptive Brownian Bridge-based Aggregation (ABBA) addresses this need by converting long numerical series into short symbolic sequences, but reductions in parameter storage and computational precision remain desirable. We propose Quantized ABBA (QABBA), a quantized version of ABBA. By quantizing the symbolic centers, QABBA reduces the parameter footprint and enables integer arithmetic while maintaining high reconstruction quality. We establish several error bounds for the additional approximation introduced by quantization: a dimension-free bound on the excess error of each segment, a time-domain reconstruction-error bound, a stability condition for symbolic assignment, and a rule for allocating bits between segment lengths and increments. The resulting symbolic strings can be passed directly to a pretrained large language model (LLM) without any extra time-series embedding layer. Experiments on the Monash regression archive, UCR Time Series Classification Archive, and UEA Multivariate Time Series Classification Archive demonstrate a practical trade-off among storage, reconstruction accuracy and downstream predictive performance. QABBA therefore provides an error-controlled, low-precision symbolic representation for time-series compression and LLM-based analysis.
Erin Carson, Xinye Chen, Fei He et al.· 0 citations
Feature-based newsvendor models use observable covariates to tailor inventory decisions, aiming to balance holding and shortage costs under demand uncertainty. However, high-dimensional feature sets often hinder interpretability and inflate data collection and implementation costs. This paper studies variable selection for the feature-based newsvendor problem under a hard cardinality constraint on the number of selected features. We formulate the resulting $\ell_0$-constrained empirical newsvendor problem with $\ell_2$-regularization, establish its computational hardness, and develop a mixed-integer second-order cone programming reformulation that strengthens the standard Big-$M$ formulation. To enable scalability beyond exact optimization, we develop a randomized-rounding algorithm with a bi-criteria guarantee and a greedy heuristic. Statistically, we provide theoretical analysis of the resulting sparse policy estimator, including finite-sample estimation error, out-of-sample risk bounds, and support recovery guarantees. Extensive experiments on both synthetic and real data illustrate the computational and statistical trade-offs among various baselines. Our results demonstrate that the proposed variable selection framework achieves competitive out-of-sample operational costs while using substantially fewer covariates.
Generative data augmentation is widely used to mitigate class imbalance, yet its theoretical effect on downstream generalization remains poorly understood. In this work, we develop a statistical framework for conditional generative augmentation and analyze its impact on classification risk. We formalize augmentation as a distribution-mixing process and show that the resulting risk distortion is controlled by both the augmentation strength and the class-conditional Wasserstein discrepancy between real and generated distributions. We further derive a capacity-dependent generalization bound based on Rademacher complexity, revealing an explicit trade-off between hypothesis complexity, augmentation intensity, and generative fidelity. Empirically, we evaluate the framework on binary and multiclass imbalanced classification tasks using Conditional GAN and Conditional WGAN-GP augmentation. Across datasets, CWGAN-GP consistently achieves lower Wasserstein discrepancies than CGAN, indicating improved distributional fidelity. However, improved fidelity does not necessarily translate into superior classification performance, with classical oversampling methods often remaining competitive. These findings support the central theoretical prediction that augmentation reliability is governed by distributional approximation error rather than predictive performance alone. Overall, this work establishes generative augmentation as a distributional perturbation process whose reliability can be quantified through Wasserstein-based measures and supported by finite-sample generalization guarantees. The proposed framework provides a principled foundation for evaluating synthetic data quality beyond classification accuracy alone.
Chathurika S Abeykoon, Mathias Nthiani Muia, Mallory Goldstein· 0 citations
At a junction, a score field can reveal weighted tangent rays, yet these first-order quantities do not determine how individual branches bend or how their densities change away from the center. Recovering this missing information is necessary for describing local continuation beyond a single point, but finite observations must separate branchwise second-order effects while allowing error in the estimated center. We address this inverse problem using matched score queries at noise scales $\sigma$ and $\lambda\sigma$. For a finite union of $C^{2,\alpha}$ half-branches in $\mathbb{R}^D$, the normalized score has the expansion $F_\sigma=F_0+\sigma G+O(\sigma^{1+\alpha})$. Matched subtraction cancels the tangent contribution and exposes $G$, which depends linearly on branchwise curvature and log-density slope. Given tangent directions and weights on distinct rays, $G$ uniquely identifies all $sD$ branch parameters, and $sD$ scalar component observations are necessary. An $O(\sigma^2)$ center error introduces $D$ translation modes, leading to $(s+1)D$ observations under full-rank calibration, except for a translation-invariant full line. We also establish a perturbation bound and a conditional kernel-density-estimation rate. Experiments reproduce the predicted population and $N^{-1/5}$ trends and remain full rank up to $D=20$ with 16 supplied branches. In end-to-end tests for $D=3$--$5$, a known-count first-order frontend yields full rank in all 135 population systems and a median relative jet error of 0.132. With strong first-order error, matched responses reduce median parameter error by a factor of 49.4 relative to naive tangent subtraction.
Reconstructing a damaged musical fragment is an inverse problem: the observed sequence contains partial information, while a raga encodes constraints limiting allowable completions. This paper formalizes a mathematical framework for this, proposing the Artificial Rosetta Stone (ARS). We separate three claims often conflated: a symbolic sequence can be reconstructed probabilistically; a sequence can be consistent with an explicit grammar; and a historical performance can be authenticated. We only support the first two. We model a raga via a finite alphabet and constraint system, using an order-k Markov model for melodic probabilities. A symmetric Dirichlet prior yields a tractable posterior. We pose missing-note reconstruction as a constrained MAP problem. For fixed-length sequences and finite-order constraints, optimization admits an exact dynamic-programming solution with worst-case time complexity $O(TN^{k+1})$. We derive the parameter count $N^k(N - 1)$, prove a concentration bound under explicit mixing assumptions, and analyze estimation error propagation. A reproducible synthetic experiment uses six raga-inspired alphabets, orders $k \in \{1, 2, 3\}$, and masking rates up to 50%. This is a proof of concept, not historical reconstruction. A real-audio feasibility pilot evaluates 30 usable sequences from 42 Yaman clips via automated pitch extraction, segmentation, and quantization. Lacking documented provenance and relying on automated transcription, this is not expert-validated archival reconstruction. Claims are tied to stated conditions, not universal properties of Hindustani music. Code: https://github.com/mathacker23/ArtificialRosettaStone.
Saanvi Raghavendran (Abstract Math Institute), Abhishek Bhattacharjee (Abstract Math Institute)· 0 citations
We consider semi-supervised classification from a partially classified sample arising from a two-component Weibull mixture. The feature is observed for all data, whereas some class labels are missing. The probability of a missing label is modelled as a function of classification uncertainty, giving a feature-dependent missing-at-random (MAR) mechanism that shares parameters with the Weibull-mixture classifier. The missing-label indicators can therefore provide information about the classifier in addition to the observed features and available class labels. Under a common Weibull shape, a Bayes' rule has at most one positive decision boundary, which is unique when the rule is nonconstant; under unequal shapes, it can have two. We characterise these decision regions, derive the Fisher information for the classifier after adjustment for nuisance parameters in the missingness model, and obtain a decision-boundary expansion of the expected error rate of the plug-in sample rule relative to the Bayes error. The expansion yields classification-specific asymptotic relative efficiency formulas for the one- and two-boundary cases and shows that a positive-definite increase in Fisher information is sufficient, but not necessary, for a smaller first-order expected error rate. Numerical studies and a semi-synthetic analysis based on hard-drive failure data illustrate potential reductions in expected error rate and improvements in decision-boundary estimation from modelling feature-dependent label missingness.
Jinran Wu, You-Gan Wang, Geoffrey J. McLachlan· 0 citations
This paper develops a general convolutional neural network (CNN) framework for detecting heterogeneous event-driven dynamics in univariate time series windows. We show that the induced CNN class exactly represents classifiers based on range, maximum drawup, maximum drawdown and slope change, and uniformly approximates realised volatility and autoregressive explosiveness on compact domains. We further establish error bounds for representative rules in finite samples and an oracle inequality for learning across them. Simulations show that the proposed model can match or outperform classifiers based on individual statistics as the training sample grows. In an application to six daily energy price series, a hierarchical CNN distinguishes event windows and event families. Applied without retraining to observations withheld after 20 February 2026, the fitted model identifies predominantly geopolitical dynamics in several oil and refined product series around the outbreak of the 2026 Iran war, while distinguishing a contemporaneous natural gas spike associated with weather.
