arXiv subjects

Jiang Hu

Publications and source records attributed to Jiang Hu.

At least 91 records · Page 5Linked to original sources

arXiv2023

No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional noncentral Sample Covariance Matrices

Let $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $, where $ \bbX_n $ is a $ p \times n $ matrix with independent standardized random variables, $ \bbR_n $ is a $ p \times n $ non-random matrix and $ \bbT_{n} $ is a $ p \times p $ non-random, nonnegative definite Hermitian matrix. The matrix $\bbB_n$ is referred to as the information-plus-noise type matrix, where $\bbR_n$ contains the information and $\bbT^{1/2}_n \bbX_n$ is the noise matrix with the covariance matrix $\bbT_{n} $. It is known that, as $ n \to \infty $, if $ p/n $ converges to a positive number, the empirical spectral distribution of $ \bbB_n $ converges almost surely to a nonrandom limit, under some mild conditions. In this paper, we prove that, under certain conditions on the eigenvalues of $ \bbR_n $ and $ \bbT_n $, for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all $ n $ sufficiently large.

math.PR↗

arXiv2023

Decentralized Riemannian natural gradient methods with Kronecker-product approximations

With a computationally efficient approximation of the second-order information, natural gradient methods have been successful in solving large-scale structured optimization problems. We study the natural gradient methods for the large-scale decentralized optimization problems on Riemannian manifolds, where the local objective function defined by the local dataset is of a log-probability type. By utilizing the structure of the Riemannian Fisher information matrix (RFIM), we present an efficient decentralized Riemannian natural gradient descent (DRNGD) method. To overcome the communication issue of the high-dimension RFIM, we consider a class of structured problems for which the RFIM can be approximated by a Kronecker product of two low-dimension matrices. By performing the communications over the Kronecker factors, a high-quality approximation of the RFIM can be obtained in a low cost. We prove that DRNGD converges to a stationary point with the best-known rate of $\mathcal{O}(1/K)$. Numerical experiments demonstrate the efficiency of our proposed method compared with the state-of-the-art ones. To the best of our knowledge, this is the first Riemannian second-order method for solving decentralized manifold optimization problems.

math.OC↗

arXiv2023

A projected semismooth Newton method for a class of nonconvex composite programs with strong prox-regularity

This paper aims to develop a Newton-type method to solve a class of nonconvex composite programs. In particular, the nonsmooth part is possibly nonconvex. To tackle the nonconvexity, we develop a notion of strong prox-regularity which is related to the singleton property, Lipschitz continuity, and monotonicity of the associated proximal operator, and we verify it in various classes of functions, including weakly convex functions, indicator functions of proximally smooth sets, and two specific sphere-related nonconvex nonsmooth functions. In this case, the problem class we are concerned with covers smooth optimization problems on manifold and certain composite optimization problems on manifold. For the latter, the proposed algorithm is the first second-order type method. Combining with the semismoothness of the proximal operator, we design a projected semismooth Newton method to find a root of the natural residual induced by the proximal gradient method. Since the corresponding natural residual may not be globally monotone, an extra projection is added on the usual semismooth Newton step and new criteria are proposed for the switching between the projected semismooth Newton step and the proximal step. The global convergence is then established under the strong prox-regularity. Based on the BD regularity condition, we establish local superlinear convergence. Numerical experiments demonstrate the effectiveness of our proposed method compared with state-of-the-art ones.

math.OC↗

arXiv2023

Sampling without replacement from a high-dimensional finite population

It is well known that most of the existing theoretical results in statistics are based on the assumption that the sample is generated with replacement from an infinite population. However, in practice, available samples are almost always collected without replacement. If the population is a finite set of real numbers, whether we can still safely use the results from samples drawn without replacement becomes an important problem. In this paper, we focus on the eigenvalues of high-dimensional sample covariance matrices generated without replacement from finite populations. Specifically, we derive the Tracy-Widom laws for their largest eigenvalues and apply these results to parallel analysis. We provide new insight into the permutation methods proposed by Buja and Eyuboglu in [Multivar Behav Res. 27(4) (1992) 509--540]. Simulation and real data studies are conducted to demonstrate our results.

math.ST↗

arXiv2022

Revisit of a Diaconis urn model

Let $G$ be a finite Abelian group of order $d$. We consider an urn in which, initially, there are labeled balls that generate the group $G$. Choosing two balls from the urn with replacement, observe their labels, and perform a group multiplication on the respective group elements to obtain a group element. Then, we put a ball labeled with that resulting element into the urn. This model was formulated by P. Diaconis while studying a group theoretic algorithm called MeatAxe (Holt and Rees (1994)). Siegmund and Yakir (2004) partially investigated this model. In this paper, we further investigate and generalize this model. More specifically, we allow a random number of balls to be drawn from the urn at each stage in the Diaconis urn model. For such a case, we verify that the normalized urn composition converges almost surely to the uniform distribution on the group $G$. Moreover, we obtain the asymptotic joint distribution of the urn composition by using the martingale central limit theorem.

math.PR↗

arXiv2022

Towards Collaborative Intelligence: Routability Estimation based on Decentralized Private Data

Applying machine learning (ML) in design flow is a popular trend in EDA with various applications from design quality predictions to optimizations. Despite its promise, which has been demonstrated in both academic researches and industrial tools, its effectiveness largely hinges on the availability of a large amount of high-quality training data. In reality, EDA developers have very limited access to the latest design data, which is owned by design companies and mostly confidential. Although one can commission ML model training to a design company, the data of a single company might be still inadequate or biased, especially for small companies. Such data availability problem is becoming the limiting constraint on future growth of ML for chip design. In this work, we propose an Federated-Learning based approach for well-studied ML applications in EDA. Our approach allows an ML model to be collaboratively trained with data from multiple clients but without explicit access to the data for respecting their data privacy. To further strengthen the results, we co-design a customized ML model FLNet and its personalization under the decentralized training scenario. Experiments on a comprehensive dataset show that collaborative training improves accuracy by 11% compared with individual local models, and our customized model FLNet significantly outperforms the best of previous routability estimators in this collaborative training flow.

cs.LG↗

arXiv2022

Spectral Statistics of Sample Block Correlation Matrices

A fundamental concept in multivariate statistics, sample correlation matrix, is often used to infer the correlation/dependence structure among random variables, when the population mean and covariance are unknown. A natural block extension of it, {\it sample block correlation matrix}, is proposed to take on the same role, when random variables are generalized to random sub-vectors. In this paper, we establish a spectral theory of the sample block correlation matrices and apply it to group independent test and related problem, under the high-dimensional setting. More specifically, we consider a random vector of dimension $p$, consisting of $k$ sub-vectors of dimension $p_t$'s, where $p_t$'s can vary from $1$ to order $p$. Our primary goal is to investigate the dependence of the $k$ sub-vectors. We construct a random matrix model called sample block correlation matrix based on $n$ samples for this purpose. The spectral statistics of the sample block correlation matrix include the classical Wilks' statistic and Schott's statistic as special cases. It turns out that the spectral statistics do not depend on the unknown population mean and covariance. Further, under the null hypothesis that the sub-vectors are independent, the limiting behavior of the spectral statistics can be described with the aid of the Free Probability Theory. Specifically, under three different settings of possibly $n$-dependent $k$ and $p_t$'s, we show that the empirical spectral distribution of the sample block correlation matrix converges to the free Poisson binomial distribution, free Poisson distribution (Marchenko-Pastur law) and free Gaussian distribution (semicircle law), respectively. We then further derive the CLTs for the linear spectral statistics of the block correlation matrix under general setting.

math.ST↗

arXiv2022

Riemannian Natural Gradient Methods

This paper studies large-scale optimization problems on Riemannian manifolds whose objective function is a finite sum of negative log-probability losses. Such problems arise in various machine learning and signal processing applications. By introducing the notion of Fisher information matrix in the manifold setting, we propose a novel Riemannian natural gradient method, which can be viewed as a natural extension of the natural gradient method from the Euclidean setting to the manifold setting. We establish the almost-sure global convergence of our proposed method under standard assumptions. Moreover, we show that if the loss function satisfies certain convexity and smoothness conditions and the input-output map satisfies a Riemannian Jacobian stability condition, then our proposed method enjoys a local linear -- or, under the Lipschitz continuity of the Riemannian Jacobian of the input-output map, even quadratic -- rate of convergence. We then prove that the Riemannian Jacobian stability condition will be satisfied by a two-layer fully connected neural network with batch normalization with high probability, provided that the width of the network is sufficiently large. This demonstrates the practical relevance of our convergence rate result. Numerical experiments on applications arising from machine learning demonstrate the advantages of the proposed method over state-of-the-art ones.

math.OC↗

arXiv2022

A CLT for the LSS of large dimensional sample covariance matrices with unbounded dispersions

In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix when the population covariance matrices are not uniformly bounded, which is a nontrivial extension of the Bai-Silverstein theorem (BST) (2004). The latter has strongly stimulated the development of high-dimensional statistics, especially the application of random matrix theory to statistics. However, the assumption of uniform boundedness of the population covariance matrices is found strongly limited to the applications of BST. The aim of this paper is to remove the blockages to the applications of BST. The new CLT, allows the spiked eigenvalues to exist and tend to infinity. It is interesting to note that the roles of either spiked eigenvalues or the bulk eigenvalues or both of the two are dominating in the CLT. Moreover, the results are checked by simulation studies with various population settings. The CLT for LSS is then applied for testing the hypothesis that a covariance matrix $ \bSi $ is equal to an identity matrix. For this, the asymptotic distributions for the corrected likelihood ratio test (LRT) and Nagao's trace test (NT) under alternative are derived, and we also propose the asymptotic power of LRT and NT under certain alternatives.

math.ST↗

arXiv2022

The limiting spectral distribution of large dimensional general information-plus-noise type matrices

Let $ X_{n} $ be $ n\times N $ random complex matrices, $R_{n}$ and $T_{n}$ be non-random complex matrices with dimensions $n\times N$ and $n\times n$, respectively. We assume that the entries of $ X_{n} $ are independent and identically distributed, $ T_{n} $ are nonnegative definite Hermitian matrices and $T_{n}R_{n}R_{n}^{*}= R_{n}R_{n}^{*}T_{n} $. The general information-plus-noise type matrices are defined by $C_{n}=\frac{1}{N}T_{n}^{\frac{1}{2}} \left( R_{n} +X_{n}\right) \left(R_{n}+X_{n}\right)^{*}T_{n}^{\frac{1}{2}} $. In this paper, we establish the limiting spectral distribution of the large dimensional general information-plus-noise type matrices $C_{n}$. Specifically, we show that as $n$ and $N$ tend to infinity proportionally, the empirical distribution of the eigenvalues of $C_{n}$ converges weakly to a non-random probability distribution, which is characterized in terms of a system of equations of its Stieltjes transform.

math.ST↗

arXiv2021

Automatic Routability Predictor Development Using Neural Architecture Search

The rise of machine learning technology inspires a boom of its applications in electronic design automation (EDA) and helps improve the degree of automation in chip designs. However, manually crafted machine learning models require extensive human expertise and tremendous engineering efforts. In this work, we leverage neural architecture search (NAS) to automate the development of high-quality neural architectures for routability prediction, which can help to guide cell placement toward routable solutions. Our search method supports various operations and highly flexible connections, leading to architectures significantly different from all previous human-crafted models. Experimental results on a large dataset demonstrate that our automatically generated neural architectures clearly outperform multiple representative manually crafted solutions. Compared to the best case of manually crafted models, NAS-generated models achieve 5.85% higher Kendall's $τ$ in predicting the number of nets with DRC violations and 2.12% better area under ROC curve (ROC-AUC) in DRC hotspot detection. Moreover, compared with human-crafted models, which easily take weeks to develop, our efficient NAS approach finishes the whole automatic search process with only 0.3 days.

cs.LG↗

arXiv2021

Toward Taming the Overhead Monster for Data-Flow Integrity

Data-Flow Integrity (DFI) is a well-known approach to effectively detecting a wide range of software attacks. However, its real-world application has been quite limited so far because of the prohibitive performance overhead it incurs. Moreover, the overhead is enormously difficult to overcome without substantially lowering the DFI criterion. In this work, an analysis is performed to understand the main factors contributing to the overhead. Accordingly, a hardware-assisted parallel approach is proposed to tackle the overhead challenge. Simulations on SPEC CPU 2006 benchmark show that the proposed approach can completely enforce the DFI defined in the original seminal work while reducing performance overhead by 4x, on average.

cs.AR↗

arXiv2021

Spiked eigenvalues of noncentral Fisher matrix with applications

In this paper, we investigate the asymptotic behavior of spiked eigenvalues of the noncentral Fisher matrix defined by ${\mathbf F}_p={\mathbf C}_n(\mathbf S_N)^{-1}$, where ${\mathbf C}_n$ is a noncentral sample covariance matrix defined by $(\mathbf Ξ+\mathbf X)(\mathbf Ξ+\mathbf X)^*/n$ and $\mathbf S_N={\mathbf Y}{\mathbf Y}^*/N$. The matrices $\mathbf X$ and $\mathbf Y$ are two independent {Gaussian} arrays, with respective $p\times n$ and $p\times N$ and the Gaussian entries of them are \textit {independent and identically distributed} (i.i.d.) with mean $0$ and variance $1$. When $p$, $n$, and $N$ grow to infinity proportionally, we establish a phase transition of the spiked eigenvalues of $\mathbf F_p$. Furthermore, we derive the \textit{central limiting theorem} (CLT) for the spiked eigenvalues of $\mathbf F_p$. As an accessory to the proof of the above results, the fluctuations of the spiked eigenvalues of ${\mathbf C}_n$ are studied, which should have its own interests. Besides, we develop the limits and CLT for the sample canonical correlation coefficients by the results of the spiked noncentral Fisher matrix and give three consistent estimators, including the population spiked eigenvalues and the population canonical correlation coefficients.

math.ST↗

arXiv2020

Net2: A Graph Attention Network Method Customized for Pre-Placement Net Length Estimation

Net length is a key proxy metric for optimizing timing and power across various stages of a standard digital design flow. However, the bulk of net length information is not available until cell placement, and hence it is a significant challenge to explicitly consider net length optimization in design stages prior to placement, such as logic synthesis. This work addresses this challenge by proposing a graph attention network method with customization, called Net2, to estimate individual net length before cell placement. Its accuracy-oriented version Net2a achieves about 15% better accuracy than several previous works in identifying both long nets and long critical paths. Its fast version Net2f is more than 1000 times faster than placement while still outperforms previous works and other neural network techniques in terms of various accuracy metrics.

cs.LG↗

arXiv2020

Fast IR Drop Estimation with Machine Learning

IR drop constraint is a fundamental requirement enforced in almost all chip designs. However, its evaluation takes a long time, and mitigation techniques for fixing violations may require numerous iterations. As such, fast and accurate IR drop prediction becomes critical for reducing design turnaround time. Recently, machine learning (ML) techniques have been actively studied for fast IR drop estimation due to their promise and success in many fields. These studies target at various design stages with different emphasis, and accordingly, different ML algorithms are adopted and customized. This paper provides a review to the latest progress in ML-based IR drop estimation techniques. It also serves as a vehicle for discussing some general challenges faced by ML applications in electronics design automation (EDA), and demonstrating how to integrate ML models with conventional techniques for the better efficiency of EDA tools.

cs.LG↗

arXiv2020

FIST: A Feature-Importance Sampling and Tree-Based Method for Automatic Design Flow Parameter Tuning

Design flow parameters are of utmost importance to chip design quality and require a painfully long time to evaluate their effects. In reality, flow parameter tuning is usually performed manually based on designers' experience in an ad hoc manner. In this work, we introduce a machine learning-based automatic parameter tuning methodology that aims to find the best design quality with a limited number of trials. Instead of merely plugging in machine learning engines, we develop clustering and approximate sampling techniques for improving tuning efficiency. The feature extraction in this method can reuse knowledge from prior designs. Furthermore, we leverage a state-of-the-art XGBoost model and propose a novel dynamic tree technique to overcome overfitting. Experimental results on benchmark circuits show that our approach achieves 25% improvement in design quality or 37% reduction in sampling cost compared to random forest method, which is the kernel of a highly cited previous work. Our approach is further validated on two industrial designs. By sampling less than 0.02% of possible parameter sets, it reduces area by 1.83% and 1.43% compared to the best solutions hand-tuned by experienced designers.

cs.LG↗

arXiv2020

PowerNet: Transferable Dynamic IR Drop Estimation via Maximum Convolutional Neural Network

IR drop is a fundamental constraint required by almost all chip designs. However, its evaluation usually takes a long time that hinders mitigation techniques for fixing its violations. In this work, we develop a fast dynamic IR drop estimation technique, named PowerNet, based on a convolutional neural network (CNN). It can handle both vector-based and vectorless IR analyses. Moreover, the proposed CNN model is general and transferable to different designs. This is in contrast to most existing machine learning (ML) approaches, where a model is applicable only to a specific design. Experimental results show that PowerNet outperforms the latest ML method by 9% in accuracy for the challenging case of vectorless IR drop and achieves a 30 times speedup compared to an accurate IR drop commercial tool. Further, a mitigation tool guided by PowerNet reduces IR drop hotspots by 26% and 31% on two industrial designs, respectively, with very limited modification on their power grids.

cs.LG↗

arXiv2020

Automatic Microprocessor Performance Bug Detection

Processor design validation and debug is a difficult and complex task, which consumes the lion's share of the design process. Design bugs that affect processor performance rather than its functionality are especially difficult to catch, particularly in new microarchitectures. This is because, unlike functional bugs, the correct processor performance of new microarchitectures on complex, long-running benchmarks is typically not deterministically known. Thus, when performance benchmarking new microarchitectures, performance teams may assume that the design is correct when the performance of the new microarchitecture exceeds that of the previous generation, despite significant performance regressions existing in the design. In this work, we present a two-stage, machine learning-based methodology that is able to detect the existence of performance bugs in microprocessors. Our results show that our best technique detects 91.5% of microprocessor core performance bugs whose average IPC impact across the studied applications is greater than 1% versus a bug-free design with zero false positives. When evaluated on memory system bugs, our technique achieves 100% detection with zero false positives. Moreover, the detection is automatic, requiring very little performance engineer time.

cs.AR↗