arXiv subjects

Annan Yu

Publications and source records attributed to Annan Yu.

17 recordsLinked to original sources

arXiv2026

Sharpen Your Flow: Sharpness-Aware Sampling for Flow Matching

Flow matching models generate samples by numerically integrating a learned velocity field, with each integration step requiring a neural network evaluation. Fast generation therefore requires using a small fixed evaluation budget effectively: the key question is not only how to integrate the flow, but where the sampler should spend its steps. We propose SharpEuler, a training-free sampler that profiles a pretrained model offline by estimating where the learned velocity field changes most rapidly along calibration trajectories. This finite-difference estimate defines a solver-aware sharpness profile, which is smoothed and converted by a quantile transform into a timestep grid for any desired inference budget. At test time, sampling remains ordinary Euler integration with the same number of model evaluations as a uniform schedule. We justify SharpEuler using three principles: a numerical principle identifying trajectory acceleration as the leading source of Euler discretization error, a variational principle deriving sharpness-based power-law timestep densities, and a statistical guarantee showing that the finite-sample calibrated sampler is stable at the terminal distribution level. Our experiments show that SharpEuler improves sample quality at fixed budgets, reducing inter-mode leakage and increasing mode coverage.

cs.LG↗

arXiv2026

Continuity Laws for Sequential Models

Inductive biases influence the behavior and performance of sequential models. In this work, we study an underexplored inductive bias in sequential modeling: continuity in time. We ask a simple question: do models motivated by continuous-time formulations, such as state-space models, actually behave continuously in time, and does this translate into better performance on tasks with continuous temporal structure? To answer this, we formalize model continuity as convergence under temporal refinement, where a model is continuous if its predictions approach an underlying continuous trajectory as the temporal discretization is refined. We show that S4 exhibits stable continuous behavior, whereas S6 (the core of Mamba) can be more sensitive to input amplitude and selective dynamics, despite being derived from a continuous dynamical system. To study whether this distinction matters for learning, we also need a corresponding notion of task continuity. We therefore introduce a metric to quantify the continuity of datasets directly from their temporal structure. Across benchmarks, we find a clear empirical alignment between task continuity, model continuity, and model performance. Beyond an inductive bias, continuity also has practical consequences: we show that it enables a simple temporal subsampling strategy that improves both efficiency and performance.

cs.LG↗

arXiv2025

HydroDiffusion: Diffusion-Based Probabilistic Streamflow Forecasting with a State Space Backbone

Recent advances have introduced diffusion models for probabilistic streamflow forecasting, demonstrating strong early flood-warning skill. However, current implementations rely on recurrent Long Short-Term Memory (LSTM) backbones and single-step training objectives, which limit their ability to capture long-range dependencies and produce coherent forecast trajectories across lead times. To address these limitations, we developed HydroDiffusion, a diffusion-based probabilistic forecasting framework with a decoder-only state space model backbone. The proposed framework jointly denoises full multi-day trajectories in a single pass, ensuring temporal coherence and mitigating error accumulation common in autoregressive prediction. HydroDiffusion is evaluated across 531 watersheds in the contiguous United States (CONUS) in the CAMELS dataset. We benchmark HydroDiffusion against two diffusion baselines with LSTM backbones, as well as the recently proposed Diffusion-based Runoff Model (DRUM). Results show that HydroDiffusion achieves strong nowcast accuracy when driven by observed meteorological forcings, and maintains consistent performance across the full simulation horizon. Moreover, HydroDiffusion delivers stronger deterministic and probabilistic forecast skill than DRUM in operational forecasting. These results establish HydroDiffusion as a robust generative modeling framework for medium-range streamflow forecasting, providing both a new modeling benchmark and a foundation for future research on probabilistic hydrologic prediction at continental scales.

cs.LG↗

arXiv2025

Understanding the Implicit Biases of Design Choices for Time Series Foundation Models

Time series foundation models (TSFMs) are a class of potentially powerful, general-purpose tools for time series forecasting and related temporal tasks, but their behavior is strongly shaped by subtle inductive biases in their design. Rather than developing a new model and claiming that it is better than existing TSFMs, e.g., by winning on existing well-established benchmarks, our objective is to understand how the various ``knobs'' of the training process affect model quality. Using a mix of theory and controlled empirical evaluation, we identify several design choices (patch size, embedding choice, training objective, etc.) and show how they lead to implicit biases in fundamental model properties (temporal behavior, geometric structure, how aggressively or not the model regresses to the mean, etc.); and we show how these biases can be intuitive or very counterintuitive, depending on properties of the model and data. We also illustrate in a case study on outlier handling how multiple biases can interact in complex ways; and we discuss implications of our results for learning the bitter lesson and building TSFMs.

cs.LG↗

arXiv2025

Understanding Transformers for Time Series: Rank Structure, Flow-of-ranks, and Compressibility

Transformers are widely used across data modalities, and yet the principles distilled from text models often transfer imperfectly to models trained to other modalities. In this paper, we analyze Transformers through the lens of rank structure. Our focus is on the time series setting, where the structural properties of the data differ remarkably from those of text or vision. We show that time-series embeddings, unlike text or vision, exhibit sharply decaying singular value spectra: small patch sizes and smooth continuous mappings concentrate the data into low-rank subspaces. From this, we prove that the associated $Q/K/V$ projections admit accurate low-rank approximations, and that attention layers become compressible in proportion to the decay of the embedding spectrum. We introduce the concept of flow-of-ranks, a phenomenon by which nonlinear mixing across depth inflates the rank, explaining why early layers are most amenable to compression and why ranks grow with depth. Guided by these theoretical and empirical results, we use these insights to compress Chronos, a large time series foundation model, achieving a reduction of $65\%$ in inference time and $81\%$ in memory, without loss of accuracy. Our findings provide principled guidance for allocating width, depth, and heads in time series foundation models, and for exploiting their inherent compressibility.

cs.LG↗

arXiv2025

Block-Biased Mamba for Long-Range Sequence Processing

Mamba extends earlier state space models (SSMs) by introducing input-dependent dynamics, and has demonstrated strong empirical performance across a range of domains, including language modeling, computer vision, and foundation models. However, a surprising weakness remains: despite being built on architectures designed for long-range dependencies, Mamba performs poorly on long-range sequential tasks. Understanding and addressing this gap is important for improving Mamba's universality and versatility. In this work, we analyze Mamba's limitations through three perspectives: expressiveness, inductive bias, and training stability. Our theoretical results show how Mamba falls short in each of these aspects compared to earlier SSMs such as S4D. To address these issues, we propose $\text{B}_2\text{S}_6$, a simple extension of Mamba's S6 unit that combines block-wise selective dynamics with a channel-specific bias. We prove that these changes equip the model with a better-suited inductive bias and improve its expressiveness and stability. Empirically, $\text{B}_2\text{S}_6$ outperforms S4 and S4D on Long-Range Arena (LRA) tasks while maintaining Mamba's performance on language modeling benchmarks.

cs.LG↗

arXiv2025

A Deep State Space Model for Rainfall-Runoff Simulations

The classical way of studying the rainfall-runoff processes in the water cycle relies on conceptual or physically-based hydrologic models. Deep learning (DL) has recently emerged as an alternative and blossomed in hydrology community for rainfall-runoff simulations. However, the decades-old Long Short-Term Memory (LSTM) network remains the benchmark for this task, outperforming newer architectures like Transformers. In this work, we propose a State Space Model (SSM), specifically the Frequency Tuned Diagonal State Space Sequence (S4D-FT) model, for rainfall-runoff simulations. The proposed S4D-FT is benchmarked against the established LSTM and a physically-based Sacramento Soil Moisture Accounting model across 531 watersheds in the contiguous United States (CONUS). Results show that S4D-FT is able to outperform the LSTM model across diverse regions. Our pioneering introduction of the S4D-FT for rainfall-runoff simulations challenges the dominance of LSTM in the hydrology community and expands the arsenal of DL tools available for hydrological modeling.

cs.LG↗

arXiv2024

Elucidating the Design Choice of Probability Paths in Flow Matching for Forecasting

Flow matching has recently emerged as a powerful paradigm for generative modeling and has been extended to probabilistic time series forecasting in latent spaces. However, the impact of the specific choice of probability path model on forecasting performance remains under-explored. In this work, we demonstrate that forecasting spatio-temporal data with flow matching is highly sensitive to the selection of the probability path model. Motivated by this insight, we propose a novel probability path model designed to improve forecasting performance. Our empirical results across various dynamical system benchmarks show that our model achieves faster convergence during training and improved predictive performance compared to existing probability path models. Importantly, our approach is efficient during inference, requiring only a few sampling steps. This makes our proposed model practical for real-world applications and opens new avenues for probabilistic forecasting.

stat.ML↗

arXiv2024

Tuning Frequency Bias of State Space Models

State space models (SSMs) leverage linear, time-invariant (LTI) systems to effectively learn sequences with long-range dependencies. By analyzing the transfer functions of LTI systems, we find that SSMs exhibit an implicit bias toward capturing low-frequency components more effectively than high-frequency ones. This behavior aligns with the broader notion of frequency bias in deep learning model training. We show that the initialization of an SSM assigns it an innate frequency bias and that training the model in a conventional way does not alter this bias. Based on our theory, we propose two mechanisms to tune frequency bias: either by scaling the initialization to tune the inborn frequency bias; or by applying a Sobolev-norm-based filter to adjust the sensitivity of the gradients to high-frequency inputs, which allows us to change the frequency bias via training. Using an image-denoising task, we empirically show that we can strengthen, weaken, or even reverse the frequency bias using both mechanisms. By tuning the frequency bias, we can also improve SSMs' performance on learning long-range sequences, averaging an 88.26% accuracy on the Long-Range Arena (LRA) benchmark tasks.

cs.LG↗

arXiv2024

HOPE for a Robust Parameterization of Long-memory State Space Models

State-space models (SSMs) that utilize linear, time-invariant (LTI) systems are known for their effectiveness in learning long sequences. To achieve state-of-the-art performance, an SSM often needs a specifically designed initialization, and the training of state matrices is on a logarithmic scale with a very small learning rate. To understand these choices from a unified perspective, we view SSMs through the lens of Hankel operator theory. Building upon it, we develop a new parameterization scheme, called HOPE, for LTI systems that utilizes Markov parameters within Hankel operators. Our approach helps improve the initialization and training stability, leading to a more robust parameterization. We efficiently implement these innovations by nonuniformly sampling the transfer functions of LTI systems, and they require fewer parameters compared to canonical SSMs. When benchmarked against HiPPO-initialized models such as S4 and S4D, an SSM parameterized by Hankel operators demonstrates improved performance on Long-Range Arena (LRA) tasks. Moreover, our new parameterization endows the SSM with non-decaying memory within a fixed time window, which is empirically corroborated by a sequential CIFAR-10 task with padded noise.

cs.LG↗

arXiv2024

How to reveal the rank of a matrix?

We study algorithms called rank-revealers that reveal a matrix's rank structure. Such algorithms form a fundamental component in matrix compression, singular value estimation, and column subset selection problems. While column-pivoted QR has been widely adopted due to its practicality, it is not always a rank-revealer. Conversely, Gaussian elimination (GE) with a pivoting strategy known as global maximum volume pivoting is guaranteed to estimate a matrix's singular values but its exponential complexity limits its interest to theory. We show that the concept of local maximum volume pivoting is a crucial and practical pivoting strategy for rank-revealers based on GE and QR. In particular, we prove that it is both necessary and sufficient; highlighting that all local solutions are nearly as good as the global one. This insight elevates Gu and Eisenstat's rank-revealing QR as an archetypal rank-revealer, and we implement a version that is observed to be at most $2\times$ more computationally expensive than CPQR. We unify the landscape of rank-revealers by considering GE and QR together and prove that the success of any pivoting strategy can be assessed by benchmarking it against a local maximum volume pivot.

math.NA↗

arXiv2023

Robustifying State-space Models for Long Sequences via Approximate Diagonalization

State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable "perturb-then-diagonalize" (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models.

cs.LG↗

arXiv2023

Leveraging the Hankel norm approximation and block-AAA algorithms in reduced order modeling

Large-scale linear, time-invariant (LTI) dynamical systems are widely used to characterize complicated physical phenomena. We propose a two-stage algorithm to reduce the order of a large-scale LTI system given samples of its transfer function for a target degree $k$ of the reduced system. In the first stage, a modified adaptive Antoulas--Anderson (AAA) algorithm is used to construct a degree $d$ rational approximation of the transfer function that corresponds to an intermediate system, which can be numerically stably reduced in the second stage using ideas from the theory on Hankel norm approximation (HNA). We also study the numerical issues of Glover's HNA algorithm and provide a remedy for its numerical instabilities. A carefully computed rational approximation of degree $d$ gives us a numerically stable algorithm for reducing an LTI system, which is more efficient than SVD-based algorithms and more accurate than moment-matching algorithms.

math.NA↗

arXiv2022

Tuning Frequency Bias in Neural Network Training with Nonuniform Data

Small generalization errors of over-parameterized neural networks (NNs) can be partially explained by the frequency biasing phenomenon, where gradient-based algorithms minimize the low-frequency misfit before reducing the high-frequency residuals. Using the Neural Tangent Kernel (NTK), one can provide a theoretically rigorous analysis for training where data are drawn from constant or piecewise-constant probability densities. Since most training data sets are not drawn from such distributions, we use the NTK model and a data-dependent quadrature rule to theoretically quantify the frequency biasing of NN training given fully nonuniform data. By replacing the loss function with a carefully selected Sobolev norm, we can further amplify, dampen, counterbalance, or reverse the intrinsic frequency biasing in NN training.

cs.LG↗

arXiv2022

On the stability of unevenly spaced samples for interpolation and quadrature

Unevenly spaced samples from a periodic function are common in signal processing and can often be viewed as a perturbed equally spaced grid. In this paper, we analyze how the uneven distribution of the samples impacts the quality of interpolation and quadrature. Starting with equally spaced nodes on $[-\pi,\pi)$ with grid spacing $h$, suppose the unevenly spaced nodes are obtained by perturbing each uniform node by an arbitrary amount $\leq \alpha h$, where $0 \leq \alpha < 1/2$ is a fixed constant. We prove a discrete version of the Kadec-1/4 theorem, which states that the nonuniform discrete Fourier transform associated with perturbed nodes has a bounded condition number independent of $h$, for any $\alpha < 1/4$. We go on to show that unevenly spaced quadrature rules converge for all continuous functions and interpolants converge uniformly for all differentiable functions whose derivative has bounded variation when $0 \leq \alpha < 1/4$. Though, quadrature rules at perturbed nodes can have negative weights for any $\alpha > 0$, we provide a bound on the absolute sum of the quadrature weights. Therefore, we show that perturbed equally spaced grids with small $\alpha$ can be used without numerical woes. While our proof techniques work primarily when $0 \leq \alpha < 1/4$, we show that a small amount of oversampling extends our results to the case when $1/4 \leq \alpha < 1/2$.

math.NA↗

arXiv2021

Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks

The standard Universal Approximation Theorem for operator neural networks (NNs) holds for arbitrary width and bounded depth. Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous nonlinear operators. In our main result, we prove that for non-polynomial activation functions that are continuously differentiable at a point with a nonzero derivative, one can construct an operator NN of width five, whose inputs are real numbers with finite decimal representations, that is arbitrarily close to any given continuous nonlinear operator. We derive an analogous result for non-affine polynomial activation functions. We also show that depth has theoretical advantages by constructing operator ReLU NNs of depth $2k^3+8$ and constant width that cannot be well-approximated by any operator ReLU NN of depth $k$, unless its width is exponential in $k$.

cs.LG↗

arXiv2020

H-chromatic symmetric functions

We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementary symmetric functions, can be realized as $H$-chromatic symmetric functions. We end with some conjectures and open problems.

math.CO↗