arXiv subjects

Jiang Hu

Publications and source records attributed to Jiang Hu.

At least 19 recordsLinked to original sources

arXiv2026

Riemannian EXTRA: Communication-efficient decentralized optimization over compact submanifolds with data heterogeneity

We consider decentralized optimization over a compact Riemannian submanifold in a network of $n$ nodes, where each node $i$ owns a smooth nonconvex local objective $f_i$ defined by its private data, and the goal is to minimize the global objective $F(x)=\frac{1}{n}\sum_{i=1}^n f_i(x)$ using only local communications. A naive Riemannian extension of EXTRA leaves the auxiliary normal-space error uncontrolled and does not inherit the Euclidean convergence guarantees. We develop projection-based Riemannian EXTRA (REXTRA) and a retraction-based counterpart that incorporate a normal correction into both the communicated variable and the auxiliary update, while using only one local-variable-sized message per optimization iteration. By jointly controlling consensus, tangent, and normal errors, our analysis interprets the update as a network-based Riemannian inexact-gradient step. For projection-based REXTRA, a sufficiently small constant stepsize and a local initialization yield vanishing consensus, stationary accumulation points, and an $\mathcal O(1/k)$ best-iterate stationarity--consensus bound. For the retraction-based method, we prove that its difference from the projection update is uniformly second order and establish the same convergence properties and rate under a local initialization. Numerical experiments on representative manifold-valued learning tasks demonstrate the communication efficiency of the REXTRA updates.

math.OC↗

arXiv2026

A counterexample to a three-step convergence bound for BFGS with exact line search

We disprove a three-step error estimate for the Broyden--Fletcher--Goldfarb--Shanno {(BFGS)} method with exact line searches. We construct a globally strongly convex function $F\in C^\infty(\mathbb R^3)$ and a single nonterminating trajectory $x_k\to0$ for which $\limsup_{k\to\infty}\|x_{k+3}\|/(\|x_{k+2}\|\|x_k\|)=+\infty$. The Hessian of $F$ lies between $\tfrac12 I$ and $\tfrac32 I$ everywhere, and $F(x)-\tfrac12\|x\|^2$ is compactly supported with all derivatives vanishing at $0$. The proof first constructs point-gradient pairs satisfying the exact BFGS step identities. A five-step cycle produces a large three-step quotient and restores an invariant plane for the actual updated matrix, allowing the cycle to be repeated with arbitrarily small gradient errors. Disjoint smooth corrections to a quadratic then realize all the pairs by one objective. Strong convexity identifies the prescribed points with the unique exact line-search minimizers. Thus smoothness and strong convexity do not imply the proposed product estimate, even for an objective that agrees with a quadratic to every order at its minimizer.

math.OC↗

arXiv2026

Formal-Verification-Guided Bayesian Optimization for Non-Intrusive Last-Mile Coverage Closure

Last-mile functional coverage closure for modern chip designs remains costly because a mature UVM environment exposes many controls while direct hits on the remaining bins are sparse. We present a non-intrusive workflow that closes pre-defined bins in an existing, human-authored functional coverage model; it does not synthesize verification objectives from the design specification. The workflow first uses abstract-syntax-tree (AST) parsing and cone-of-influence (COI) tracing to map each residual target to existing testbench knobs. Under a matched 20-simulation budget, Bayesian optimization (BO) is compared with continued constrained-random verification (CRV) and random search over the same AST/COI-reduced space. Formal reachability then removes only targets supported by complete unreachability proofs, after which retained targets are grouped by formal difficulty and processed by group-specific BO under a second matched 20-simulation budget. Cleanup and grouping never add numerator bins; only simulator coverage artifacts do. On OpenTitan AES, first-pass reduced-space BO reaches 241/271 bins; formal cleanup removes two denominator bins; the integrated post-cleanup stage - difficulty grouping, policy routing, and group-specific BO - adds 20 simulator-confirmed hits, yielding 261/271 raw coverage (96.31%) and 261/269 effective coverage (97.03%). On Entropy Source, first-pass reduced-space BO reaches 2117/2277 bins; formal cleanup removes ten denominator bins; the integrated post-cleanup stage - difficulty grouping, policy routing, and group-specific BO - adds 93 simulator-confirmed hits, yielding 2210/2277 raw coverage (97.06%) and 2210/2267 effective coverage (97.49%). These results characterize the integrated diagnosis-and-search workflow; the present evaluation does not isolate the causal contributions of difficulty classification, policy routing, and post-cleanup BO.

cs.AR↗

arXiv2026

PLADOS: A Parameter-free Landing Algorithm for Decentralized Optimization on the Stiefel Manifold

Decentralized optimization on the Stiefel manifold has broad applications in machine learning and signal processing. We propose PLADOS, a parameter-free landing algorithm for decentralized optimization on the Stiefel manifold. Each iteration combines local stochastic gradient updates with a single retraction-free communication step over the communication graph. A key convex-like property of the intersection of the nonconvex Stiefel manifold and consensus constraints is characterized through a restricted secant inequality for the penalty-only scheme, ensuring local contraction of the manifold consensus error without introducing additional parameters that require tuning. For $n$ nodes using independent local batches of size $b$, we prove that PLADOS achieves linear speedup with respect to the number of nodes with an asymptotic convergence rate of $O(σ/\sqrt{nbK}+(nξ^2)^{1/3}/K^{2/3})$ after $K$ iterations, where $σ^2>0$ is the sampling variance and $ξ^2$ quantifies gradient heterogeneity. Experiments demonstrate stability across tested stepsizes and networks, time efficiency, and linear speedup with respect to the number of nodes.

math.OC↗

arXiv2026

Stochastic Subgradient Descent at Sharply Repulsive Points: A Bounded Noise Counterexample and Gaussian Noise Avoidance

Motivated by the open question of Bianchi, Hachem, and Schechtman (Bianchi et al., 2024, Remark 4), we show that in the non-weakly convex definable setting, omnidirectional noise with conditional fourth-moment control does not suffice for universal almost-sure avoidance of sharply repulsive critical points. We demonstrate this failure by constructing a globally Lipschitz, coercive, definable objective that is not weakly convex. For this objective and its sufficiently small linear perturbations, stochastic subgradient descent with independent noise uniformly distributed on a ball in $\mathbb{R}^2$ converges with probability $1$ to a nonminimal sharply repulsive critical point from every initial point in a specified ball. A two-cycle argument establishes uniform bounds on the rescaled iterates, yielding convergence to the sharply repulsive critical point. In contrast, for locally Lipschitz definable objectives, Gaussian noise ensures simultaneous almost-sure avoidance of any finite collection of sharply repulsive points. Through a construction of difference quotient functions, we show that convergence to any such point with positive probability would produce a stationary distribution with expected descent $0$, whereas a global positive lower bound on the Fréchet subgradient norms of the limiting functions forces strictly positive expected descent under the same distribution. This contradiction proves avoidance, with the everywhere positive Gaussian density playing a key role.

math.OC↗

arXiv2026

Edgeworth corrections for the spiked eigenvalues of non-Gaussian sample covariance matrices with applications

Yang and Johnstone (2018) established an Edgeworth correction for the largest sample eigenvalue in a spiked covariance model under the assumption of Gaussian observations, leaving the extension to non-Gaussian settings as an open problem. In this paper, we address this issue by establishing first-order Edgeworth expansions for spiked eigenvalues in both single-spike and multi-spike scenarios with non-Gaussian data. Leveraging these expansions, we construct more accurate confidence intervals for the population spiked eigenvalues and propose a novel estimator for the number of spikes. Simulation studies demonstrate that our proposed methodology outperforms existing approaches in both robustness and accuracy across a wide range of settings, particularly in low-dimensional cases.

math.ST↗

arXiv2026

A Human-AI Collaborative Workflow for Mathematical Discovery: A Case Study in Grover-Compatible Riemannian Optimization

We investigate how large language models can be used as research tools in scientific computing while preserving mathematical rigor. We propose a human-in-the-loop workflow for interactive theorem proving and discovery with LLMs. Human experts retain control over problem formulation and assumptions, while the model searches for proofs or contradictions, proposes candidate properties and theorems, and helps construct structures and parameters that satisfy explicit constraints, supported by numerical experiments and simple verification checks. Experts treat these outputs as raw material, further refine them, and organize the results into precise statements and rigorous proofs. We instantiate this workflow in a main case study on the connection between manifold optimization and Grover's quantum search algorithm, where the pipeline identifies invariant subspaces and explores Grover-compatible retractions. The main case study uses the corresponding Grover-compatible convergence analysis, including an $O(\sqrt{N} \log(1/\varepsilon))$ PL-based bound established in the companion mathematical work, to illustrate the refinement stage of the workflow. Prompt records and reusable templates for implementing the workflow are provided. We further include a multi-oracle case study, document representative failed and corrected routes arising from this setting, and provide a structured failure-mode analysis.

cs.HC↗

arXiv2026

Local Laws and Edge Universality for Noncentral Sample Covariance Matrices

We consider the real noncentral sample covariance matrices $\mathcal{W}=YY^\top$ with $Y=A+Σ^{1/2}X$. Here $A\in\mathbb{R}^{M\times N}$ is deterministic, $Σ\in \mathbb{R}^{M\times M}$ is a deterministic positive definite population covariance matrix and $X\in\mathbb{R}^{M\times N}$ has independent centered entries with variance $N^{-1}$. We prove local laws near regular right edges down to optimal spectral scales without requiring the commutativity of $AA^\top$ and $Σ$. As a consequence, we obtain optimal eigenvalue rigidity at the rightmost regular edge and delocalization of the corresponding left and right singular vectors. We also show that, with high probability, there are no eigenvalues in the adjacent spectral gap beyond the optimal $N^{-2/3}$ edge scale, up to an arbitrarily small $N^\varepsilon$ loss. Finally, we establish edge universality at the rightmost regular edge: after centering and scaling, the largest eigenvalue converges to the Tracy--Widom distribution. The main technical ingredient is a stability analysis of the matrix Dyson equation (MDE) associated with the linearization of $Y$, whose self-energy operator does not satisfy the flatness condition of the general MDE theory. Exploiting the special block structure, we reduce the stability analysis exactly to a two-dimensional operator. This reduction yields regularity of the spectral density and square-root behavior at regular right edges, together with sharp stability bounds near such edges.

math.PR↗

arXiv2026

Decentralized projected Riemannian stochastic recursive momentum method for nonconvex optimization

This paper studies decentralized optimization over a compact submanifold within a communication network of $n$ nodes, where each node possesses a smooth non-convex local cost function, and the goal is to jointly minimize the sum of these local costs. We focus particularly on the online setting, where local data is processed in real-time as it streams in, without the need for full data storage. We propose a decentralized projected Riemannian stochastic recursive momentum (DPRSRM) method that employs local hybrid stochastic gradient estimators and uses the network to track the global gradient. DPRSRM achieves an oracle complexity of $\mathcal{O}(ε^{-\frac{3}{2}})$, outperforming existing methods that have at most $\mathcal{O}(ε^{-2})$ complexity. Our method requires only $\mathcal{O}(1)$ gradient evaluations per iteration for each local node and does not require restarting with a large batch gradient. Furthermore, we demonstrate the effectiveness of our proposed methods compared to state-of-the-art ones through numerical experiments on principal component analysis problems and low-rank matrix completion.

math.OC↗

arXiv2026

ReasFlow: Assisting Reasoning-Centric Scientific Discovery in Applied Mathematics via a Knowledge-Based Multi-Agent System

Recent advances in Large Language Models have fueled autonomous AI agents capable of tackling complex scientific tasks, yet existing automated research systems remain predominantly focused on empirically driven domains with quantitative benchmarks, leaving theory-driven discovery, particularly in mathematically grounded disciplines requiring rigorous proofs and synthesis of domain knowledge, largely underexplored. Key challenges include the difficulty of verifying theoretical reasoning at scale, insufficient reasoning ability for autonomous frontier exploration, and a scarcity of procedural heuristics in the literature. We introduce ReasFlow, an end-to-end autonomous agent system for reasoning-centric scientific discovery that operationalizes a collaborative paradigm where the human expert acts as Principal Investigator while the agent executes rigorous derivations as a capable graduate student. ReasFlow incorporates (i) a robust internal verification loop that audits logical coherence and corrects fundamental errors prior to human inspection, and (ii) an automated knowledge retrieval and self-improvement mechanism that proactively surfaces both declarative facts and overlooked procedural heuristics, substantially reducing expert intervention. The system unifies literature synthesis, algorithm design, theorem proving, experimentation, and manuscript preparation in a single system. Deployed to autonomously generate five complete research papers with rigorous theoretical and empirical content from minimal prompts, ReasFlow consistently achieves the highest evaluation scores among state-of-the-art open-access baselines under a curated LLM-based review rubric. ReasFlow is publicly accessible via the ReasLab platform, providing a collaborative workspace for AI-assisted theoretical research. Github repo: https://github.com/reaslab/ReasFlow.git.

cs.AI↗

arXiv2026

Distributed Online Estimation of Spiked Eigenvalues with Adaptive Weighting under Persistent Aspect Ratio Heterogeneity

We study online estimation of spiked covariance eigenvalues from observations distributed across $L$ nodes with heterogeneous and persistent effective sample sizes. In the proportional high-dimensional regime, local Rayleigh statistics are deterministically distorted by node-specific aspect ratios $c_{\ell,t}=p/N^{\mathrm{eff}}_{\ell,t}$, and direct aggregation of uncorrected statistics converges to the wrong limit. We propose a correct-then-aggregate framework in which each node removes its deterministic bias via an inverse Rayleigh transfer map, and the server fuses corrected estimates using adaptive soft-max weights based on predictable fluctuation metrics, transmitting only $O(k)$ scalars per active node per round. We establish consistency and asymptotic normality of the global estimator, enabling valid online inference, and derive non-asymptotic bounds quantifying how accuracy improves with the number of nodes and their effective sample sizes. The adaptive weights achieve variance reduction comparable to oracle inverse-variance weighting, confirming the data-driven construction is nearly efficient. Simulation studies validate these properties. An application to cross-venue monitoring of a dominant market factor shows the method tracks systemic risk in real time while substantially reducing communication cost relative to a centralized pooled approach.

math.ST↗

arXiv2026

Incremental Aggregation on the Grassmannian for Asynchronous Eigenspace Computation

We study asynchronous optimization for finite-sum eigenspace computation in heterogeneous distributed systems. The theoretical foundations for asynchronous eigenspace computation remain scarce, with existing approaches offering limited coverage of dynamics directly on the Grassmannian under stale information. In this paper, we propose a Grassmannian incremental aggregation method that refreshes only arriving components and reuses cached gradients, retaining low per-update cost without global synchronization. The method employs an extrinsic polar update that preserves the intrinsic subspace geometry without requiring parallel transport of stale tangent vectors. Our analysis establishes a tight angle-dependent gradient-dominance characterization of the objective and a basin-invariance property for stale aggregated updates. These yield two-phase linear convergence, comprising an explicit broad-basin regime and a sharper local regime, with constants controlled by component spectral spreads. Experiments on serial and distributed PCA demonstrate improved sample efficiency and wall-clock convergence over representative baselines.

math.OC↗

arXiv2026

A Systolic Array Architecture for Nonlinear Activation Functions and Softmax Computation using Chebyshev Polynomials

Neural Network Accelerators have gained popularity in recent years due to their greater efficiency than CPU-based platforms. Often, these accelerators utilize different hardware units for univariate activation functions, such as tanh, and the multivariate softmax, thereby missing opportunities for resource sharing between them. In this paper, we describe a novel systolic array-based activation unit architecture that supports multiple univariate activation functions as well as the softmax function. By utilizing Chebyshev polynomial approximations, our activation function unit achieves up to 71% lower mean absolute error for tanh compared to a CORDIC baseline, while using 4.6% less area and 5.1% less power. Our softmax approximation enables a 44.6% and 79.0% lower KL divergence compared to CORDIC and a piecewise-linear approximation, respectively.

cs.AR↗

arXiv2026

EDATracer: An Agentic Framework for Large-Scale EDA Artifact Analysis

Modern chip design relies on electronic design automation (EDA) tools that generate large, heterogeneous artifacts, including source files, scripts, logs, netlists, and reports. Analyzing these artifacts is critical for debugging, optimization, and design-flow understanding, but remains difficult because relevant evidence is often distributed across many artifact types and design stages. Although LLM agents show promise for EDA assistance, existing approaches lack public benchmarks for large-scale cross-artifact analysis and often struggle to ground reasoning in tool-generated evidence. We present EDATracer, an agentic framework for evidence-grounded EDA artifact analysis. EDATracer organizes design artifacts into a knowledge graph paired with a semantic vector index, enabling LLM agents to retrieve evidence across source files, logs, netlists, and reports. We curate an 18.9 GB dataset of 2,787 synthesizable open-source chip designs and introduce a 90-question benchmark spanning factual, statistical, and reasoning tasks. Across evaluated agents, EDATracer achieves the best pass@1 accuracy, outperforming Cursor and Claude Code by 6.4 and 7.2 percentage points on average, while using 2.0-3.2x fewer tokens.

cs.AR↗

arXiv2026

Retraction-Free Optimization over the Stiefel Manifold for the LoRA Fine-Tuning

Optimization over the Stiefel manifold plays a significant role in various machine learning tasks. Existing methods either use the retraction operators, requiring costly orthonormalization for large-scale matrices, or employ landing methods that rely on careful step size selection and penalty parameter tuning. To address these challenges, we propose a retraction-free and penalty parameter-free algorithm that directly lands on the manifold. By leveraging the strongly-convex-like property of the quadratic penalty function and the proximal smoothness of the Stiefel manifold, we establish global convergence guarantees with the best-known iteration complexities under both constant and diminishing step sizes. Then, we reformulate the low-rank adaptation (LoRA) fine-tuning problem for large language models as a manifold optimization problem, introducing Manifold-LoRA for geometry-accelerated adaptation. This approach employs the proposed landing technique and a carefully designed step size strategy to accelerate the training process. Numerical experiments on benchmark datasets demonstrate the efficiency and strong downstream performance of the proposed method.

cs.LG↗

arXiv2026

Transfer Learning for Linear Discriminant Analysis with a Shared Classification Signal

This paper studies transfer learning for linear discriminant analysis in high-dimensional two-class classification. We consider one target domain and several source domains, where the mean difference in each domain is decomposed into a deterministic common component and a domain-specific random deviation. The common component represents a shared classification signal across domains, while the random deviation captures domain-specific heterogeneity. Under spiked covariance models, we derive deterministic limits for the target-domain Gaussian-calibrated error of weighted transfer classifiers under both homogeneous and heterogeneous covariance settings. These limits quantify the effects of the shared signal, domain-specific variation, dimension-to-sample-size ratios, and spike structures on transfer performance. They further lead to oracle transfer weights and consistent data-driven plug-in estimators. We also characterize the intercept bias induced by unbalanced target-domain class sample sizes and provide an asymptotically optimal correction.

stat.ME↗

arXiv2026

Achieving double-logarithmic precision dependence in optimization-based quantum unstructured search

Grover's algorithm is a fundamental quantum algorithm that achieves a quadratic speedup for unstructured search problems of size $N$. Recent studies have reformulated this task as a maximization problem on the unitary manifold and solved it via linearly convergent Riemannian gradient ascent (RGA) methods, resulting in a complexity of $O(\sqrt{N/M}\log (1/\varepsilon))$, where $M$ denotes the number of target items and $\varepsilon$ denotes the success probability error. In this work, we adopt the Riemannian modified Newton (RMN) method to solve the quantum search problem, under the assumption that the ratio $ M/N$ is known. We show that, in this setting, the Riemannian Newton direction is collinear with the Riemannian gradient in the sense that the Riemannian gradient is always an eigenvector of the corresponding Riemannian Hessian. This structure removes the overhead of Hessian inversion and allows the proposed RMN method to retain the local quadratic convergence in terms of the error $\varepsilon$. More precisely, we rigorously prove an overall complexity of $O(\sqrt{N/M}+\log\log(1/\varepsilon))$. Furthermore, our approach remains Grover-compatible, namely, it relies exclusively on the standard Grover diffusion and oracle operators to ensure algorithmic implementability, and its parameter update process can be efficiently precomputed on classical computers.

quant-ph↗

arXiv2026

A Grover-compatible manifold optimization algorithm for quantum search

Grover's algorithm is a fundamental quantum algorithm that offers a quadratic speedup for the unstructured search problem by alternately applying physically implementable oracle and diffusion operators. In this paper, we reformulate the unstructured search as a maximization problem on the unitary manifold and solve it via the Riemannian gradient ascent (RGA) method. To overcome the difficulty that generic RGA updates do not, in general, correspond to physically implementable quantum operators, we introduce Grover-compatible retractions to restrict RGA updates to valid oracle and diffusion operators. Theoretically, we establish a local Riemannian $μ$-Polyak-Łojasiewicz (PL) inequality with $μ= \tfrac{1}{2}$, which yields a linear convergence rate of $1 - κ^{-1}$ toward the global solution. Here, the condition number $κ= L_{\mathrm{Rie}} / μ$, where $L_{\mathrm{Rie}}$ denotes the Riemannian Lipschitz constant of the gradient. Taking into account both the geometry of the unitary manifold and the special structure of the cost function, we show that $L_{\mathrm{Rie}} = O(\sqrt{N})$ for problem size $N = 2^n$. Consequently, the resulting iteration complexity is $O(\sqrt{N} \log(1/\varepsilon))$ for attaining an $\varepsilon$-accurate solution, which matches the quadratic speedup of $O(\sqrt{N})$ achieved by Grover's algorithm. These results demonstrate that an optimization-based viewpoint can offer fresh conceptual insights and lead to new advances in the design of quantum algorithms.

quant-ph↗