arXiv subjects

Ke Ye

Publications and source records attributed to Ke Ye.

At least 55 records · Page 3Linked to original sources

arXiv2024

Simple matrix models for the flag, Grassmann, and Stiefel manifolds

We derive three families of orthogonally-equivariant matrix submanifold models for the Grassmann, flag, and Stiefel manifolds respectively. These families are exhaustive -- every orthogonally-equivariant submanifold model of the lowest dimension for any of these manifolds is necessarily a member of the respective family, with a small number of exceptions. They have several computationally desirable features. The orthogonal equivariance allows one to obtain, for various differential geometric objects and operations, closed-form analytic expressions that are readily computable with standard numerical linear algebra. The minimal dimension aspect translates directly to a speed advantage in computations. And having an exhaustive list of all possible matrix models permits one to identify the model with the lowest matrix condition number, which translates to an accuracy advantage in computations. As an interesting aside, we will see that the family of models for the Stiefel manifold is naturally parameterized by the Cartan manifold, i.e., the positive definite cone equipped with its natural Riemannian metric.

math.NA↗

arXiv2024

Minimal equivariant embeddings of the Grassmannian and flag manifold

We show that the flag manifold $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$, with Grassmannian the special case $p=1$, has an $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding in an Euclidean space of dimension $(n-1)(n+2)/2$, two orders of magnitude below the current best known result. We will show that the value $(n-1)(n+2)/2$ is the smallest possible and that any $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding of $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$ in an ambient space of minimal dimension is equivariantly equivalent to the aforementioned one.

math.RT↗

arXiv2024

A quasi-optimal lower bound for skew polynomial multiplication

We establish a lower bound for the complexity of multiplying two skew polynomials. The lower bound coincides with the upper bound conjectured by Caruso and Borgne in 2017, up to a log factor. We present algorithms for three special cases, indicating that the aforementioned lower bound is quasi-optimal. In fact, our lower bound is quasi-optimal in the sense of bilinear complexity. In addition, we discuss the average bilinear complexity of simultaneous multiplication of skew polynomials and the complexity of skew polynomial multiplication in the case of towers of extensions.

cs.CC↗

arXiv2024

Intrinsic Gaussian Processes on Manifolds and Their Accelerations by Symmetry

Amidst the growing interest in nonparametric regression, we address a significant challenge in Gaussian processes(GP) applied to manifold-based predictors. Existing methods primarily focus on low dimensional constrained domains for heat kernel estimation, limiting their effectiveness in higher-dimensional manifolds. Our research proposes an intrinsic approach for constructing GP on general manifolds such as orthogonal groups, unitary groups, Stiefel manifolds and Grassmannian manifolds. Our methodology estimates the heat kernel by simulating Brownian motion sample paths using the exponential map, ensuring independence from the manifold's embedding. The introduction of our strip algorithm, tailored for manifolds with extra symmetries, and the ball algorithm, designed for arbitrary manifolds, constitutes our significant contribution. Both algorithms are rigorously substantiated through theoretical proofs and numerical testing, with the strip algorithm showcasing remarkable efficiency gains over traditional methods. This intrinsic approach delivers several key advantages, including applicability to high dimensional manifolds, eliminating the requirement for global parametrization or embedding. We demonstrate its practicality through regression case studies (torus knots and eight dimensional projective spaces) and by developing binary classifiers for real world datasets (gorilla skulls planar images and diffusion tensor images). These classifiers outperform traditional methods, particularly in limited data scenarios.

math.OC↗

arXiv2024

SpacTor-T5: Pre-training T5 Models with Span Corruption and Replaced Token Detection

Pre-training large language models is known to be extremely resource intensive and often times inefficient, under-utilizing the information encapsulated in the training text sequences. In this paper, we present SpacTor, a new training procedure consisting of (1) a hybrid objective combining span corruption (SC) and token replacement detection (RTD), and (2) a two-stage curriculum that optimizes the hybrid objective over the initial $τ$ iterations, then transitions to standard SC loss. We show empirically that the effectiveness of the hybrid objective is tied to the two-stage pre-training schedule, and provide extensive analysis on why this is the case. In our experiments with encoder-decoder architectures (T5) on a variety of NLP tasks, SpacTor-T5 yields the same downstream performance as standard SC pre-training, while enabling a 50% reduction in pre-training iterations and 40% reduction in total FLOPs. Alternatively, given the same amount of computing budget, we find that SpacTor results in significantly improved downstream benchmark performance.

cs.LG↗

arXiv2023

Bundle-based similarity measurement for positive semidefinite matrices

Positive semidefinite (PSD) matrices are indispensable in many fields of science. A similarity measurement for such matrices is usually an essential ingredient in the mathematical modelling of a scientific problem. This paper proposes a unified framework to construct similarity measurements for PSD matrices. The framework is obtained by exploring the fiber bundle structure of the cone of PSD matrices and generalizing the idea of the point-set distance previously developed for linear subsapces and positive definite (PD) matrices. The framework demonstrates both theoretical advantages and computational convenience: (1) We prove that the similarity measurement constructed by the framework can be recognized either as the cost of a parallel transport or as the length of a quasi-geodesic curve. (2) We extend commonly used divergences for equidimensional PD matrices to the non-equidimensional case. Examples include Kullback-Leibler divergence, Bhattacharyya divergence and Rényi divergence. We prove that these extensions enjoy the same consistency property as their counterpart for geodesic distance. (3) We apply our geometric framework to further extend those in (2) to similarity measurements for arbitrary PSD matrices. We also provide simple formulae to compute these similarity measurements in most situations.

math.NA↗

arXiv2023

FFINet: Future Feedback Interaction Network for Motion Forecasting

Motion forecasting plays a crucial role in autonomous driving, with the aim of predicting the future reasonable motions of traffic agents. Most existing methods mainly model the historical interactions between agents and the environment, and predict multi-modal trajectories in a feedforward process, ignoring potential trajectory changes caused by future interactions between agents. In this paper, we propose a novel Future Feedback Interaction Network (FFINet) to aggregate features the current observations and potential future interactions for trajectory prediction. Firstly, we employ different spatial-temporal encoders to embed the decomposed position vectors and the current position of each scene, providing rich features for the subsequent cross-temporal aggregation. Secondly, the relative interaction and cross-temporal aggregation strategies are sequentially adopted to integrate features in the current fusion module, observation interaction module, future feedback module and global fusion module, in which the future feedback module can enable the understanding of pre-action by feeding the influence of preview information to feedforward prediction. Thirdly, the comprehensive interaction features are further fed into final predictor to generate the joint predicted trajectories of multiple agents. Extensive experimental results show that our FFINet achieves the state-of-the-art performance on Argoverse 1 and Argoverse 2 motion forecasting benchmarks.

cs.CV↗

arXiv2023

Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time

The problem of verifying the nonnegativity of a function on a finite abelian group is a long-standing challenging problem. The basic theory of representation theory of finite groups indicates that a function $f$ on a finite abelian group $G$ can be written as a linear combination of characters of irreducible representations of $G$ by $ f(x)=\sum_{χ\in \widehat{G}} \widehat{f} (χ)χ(x)$, where $\widehat{G}$ is the dual group of $G$ consisting of all characters of $G$ and $ \widehat{f} (χ)$ is the Fourier coefficient of $f$ at $χ\in \widehat{G}$. In this paper, we show that by performing the fast (inverse) Fourier transform, we are able to compute a sparse Fourier sum of squares (FSOS) certificate of $f$ on a finite abelian group $G$ with complexity \if $\operatorname{O}\left(|G| \log(|G|)+\log(k_{\min})\operatorname{SDP}(2k_{\min})\right)$,\fi that is quasi-linear in the order of $G$ and polynomial in the FSOS sparsity \if $k_{\min}$\fi of $f$. Moreover, for a nonnegatvie function $f$ on a finite abelian group $G$ and a set $S \subset \widehat{G}$, we give a lower bound of the constant $M$ such that $f+M$ admits an FSOS supported on} $S$. We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of orders up to $10^7$. As applications, we also solve some combinatorial optimization problems and the sum of Hermitian squares (SOHS) problem \if on $\mathbb{T}^n$\fi by sparse FSOS.

math.OC↗

arXiv2023

Complex matrix inversion via real matrix inversions

We study the inversion analog of the well-known Gauss algorithm for multiplying complex matrices. A simple version is $(A + iB)^{-1} = (A + BA^{-1}B)^{-1} - i A^{-1}B(A+BA^{-1} B)^{-1}$ when $A$ is invertible, which may be traced back to Frobenius but has received scant attention. We prove that it is optimal, requiring fewest matrix multiplications and inversions over the base field, and we extend it in three ways: (i) to any invertible $A + iB$ without requiring $A$ or $B$ be invertible; (ii) to any iterated quadratic extension fields, with $\mathbb{C}$ over $\mathbb{R}$ a special case; (iii) to Hermitian positive definite matrices $A + iB$ by exploiting symmetric positive definiteness of $A$ and $A + BA^{-1}B$. We call all such algorithms Frobenius inversions, which we will see do not follow from Sherman--Morrison--Woodbury type identities and cannot be extended to Moore--Penrose pseudoinverse. We show that a complex matrix with well-conditioned real and imaginary parts can be arbitrarily ill-conditioned, a situation tailor-made for Frobenius inversion. We prove that Frobenius inversion for complex matrices is faster than standard inversion by LU decomposition and Frobenius inversion for Hermitian positive definite matrices is faster than standard inversion by Cholesky decomposition. We provide extensive numerical experiments, applying Frobenius inversion to solve linear systems, evaluate matrix sign function, solve Sylvester equation, and compute polar decomposition, showing that Frobenius inversion can be more efficient than LU/Cholesky decomposition with negligible loss in accuracy. A side result is a generalization of Gauss multiplication to iterated quadratic extensions, which we show is intimately related to the Karatsuba algorithm for fast integer multiplication and multidimensional fast Fourier transform.

math.NA↗

arXiv2023

The Lazy Neuron Phenomenon: On Emergence of Activation Sparsity in Transformers

This paper studies the curious phenomenon for machine learning models with Transformer architectures that their activation maps are sparse. By activation map we refer to the intermediate output of the multi-layer perceptrons (MLPs) after a ReLU activation function, and by sparse we mean that on average very few entries (e.g., 3.0% for T5-Base and 6.3% for ViT-B16) are nonzero for each input to MLP. Moreover, larger Transformers with more layers and wider MLP hidden dimensions are sparser as measured by the percentage of nonzero entries. Through extensive experiments we demonstrate that the emergence of sparsity is a prevalent phenomenon that occurs for both natural language processing and vision tasks, on both training and evaluation data, for Transformers of various configurations, at layers of all depth levels, as well as for other architectures including MLP-mixers and 2-layer MLPs. We show that sparsity also emerges using training datasets with random labels, or with random inputs, or with infinite amount of data, demonstrating that sparsity is not a result of a specific family of datasets. We discuss how sparsity immediately implies a way to significantly reduce the FLOP count and improve efficiency for Transformers. Moreover, we demonstrate perhaps surprisingly that enforcing an even sparser activation via Top-k thresholding with a small value of k brings a collection of desired but missing properties for Transformers, namely less sensitivity to noisy training data, more robustness to input corruptions, and better calibration for their prediction confidence.

cs.LG↗

arXiv2023

Generalizing Frobenius Inversion to Quaternion Matrices

In this paper we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient that other existing methods. Moreover, our algorithm is optimal in the sense of the least number of complex inversions. On the practice side, our algorithm outperforms existing algorithms on randomly generated matrices. We argue that this algorithm can be used to improve the practical utility of recursive Strassen-type algorithms by providing the fastest possible base case for the recursive decomposition process when applied to quaternion matrices.

math.NA↗

arXiv2023

Lower Bounds of Functions on Finite Abelian Groups

The problem of computing the optimum of a function on a finite set is an important problem in mathematics and computer science. Many combinatorial problems such as MAX-SAT and MAXCUT can be recognized as optimization problems on the hypercube $C_2^n = \{-1,1\}^n$ consisting of $2^n$ points. It has been noticed that if a finite set is equipped with an abelian group structure, then one can efficiently certify nonnegative functions on it by Fourier sum of squares (FSOS). Motivated by these works, this paper is devoted to developing a framework to find a lower bound of a function on a finite abelian group efficiently. We implement our algorithm by the SDP solver SDPNAL+ for computing the verified lower bound of $f$ on $G$ and test it on the MAX-2SAT and MAX-3SAT benchmark problems from the Max-SAT competitions in 2009 and 2016. Beyond that, we also test our algorithm on random functions on $C_3^n$. These experiments demonstrate the advantage of our algorithm over previously known methods.

math.OC↗

arXiv2022

Simpler flag optimization

We study the geometry of flag manifolds under different embeddings into a product of Grassmannians. We show that differential geometric objects and operations -- tangent vector, metric, normal vector, exponential map, geodesic, parallel transport, gradient, Hessian, etc -- have closed-form analytic expressions that are computable with standard numerical linear algebra. Furthermore, we are able to derive a coordinate descent method in the flag manifold that performs well compared to other gradient descent methods.

math.OC↗

arXiv2022

Skew-sparse matrix multiplication

Based on the observation that $\mathbb{Q}^{(p-1) \times (p-1)}$ is isomorphic to a quotient skew polynomial ring, we propose a new method for $(p-1)\times (p-1)$ matrix multiplication over $\mathbb{Q}$, where $p$ is a prime number. The main feature of our method is the acceleration for matrix multiplication if the product is skew-sparse. Based on the new method, we design a deterministic algorithm with complexity $O(T^{ω-2} p^2)$, where $T\le p-1$ is a parameter determined by the skew-sparsity of input matrices and $ω$ is the asymptotic exponent of matrix multiplication. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity $O^\thicksim(t^{ω-2}p^2+p^2\log\frac{1}ν)$, where $t\le p-1$ is the skew-sparsity of the product and $ν$ is the probability parameter.

cs.CC↗

arXiv2022

When geometry meets optimization theory: partially orthogonal tensors

Due to the multi-linearity of tensors, most algorithms for tensor optimization problems are designed based on the block coordinate descent method. Such algorithms are widely employed by practitioners for their implementability and effectiveness. However, these algorithms usually suffer from the lack of theoretical guarantee of global convergence and analysis of convergence rate. In this paper, we propose a block coordinate descent type algorithm for the low rank partially orthogonal tensor approximation problem and analyse its convergence behaviour. To achieve this, we carefully investigate the variety of low rank partially orthogonal tensors and its geometric properties related to the parameter space, which enable us to locate KKT points of the concerned optimization problem. With the aid of these geometric properties, we prove without any assumption that: (1) Our algorithm converges globally to a KKT point; (2) For any given tensor, the algorithm exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual $O(1/k)$ for first order methods in nonconvex optimization; {(3)} For a generic tensor, our algorithm converges $R$-linearly.

math.OC↗

arXiv2020

Simpler Grassmannian optimization

There are two widely used models for the Grassmannian $\operatorname{Gr}(k,n)$, as the set of equivalence classes of orthogonal matrices $\operatorname{O}(n)/(\operatorname{O}(k) \times \operatorname{O}(n-k))$, and as the set of trace-$k$ projection matrices $\{P \in \mathbb{R}^{n \times n} : P^{\mathsf{T}} = P = P^2,\; \operatorname{tr}(P) = k\}$. The former, standard in manifold optimization, has the advantage of giving numerically stable algorithms but the disadvantage of having to work with equivalence classes of matrices. The latter, widely used in coding theory and probability, has the advantage of using actual matrices (as opposed to equivalence classes) but working with projection matrices is numerically unstable. We present an alternative that has both advantages and suffers from neither of the disadvantages; by representing $k$-dimensional subspaces as symmetric orthogonal matrices of trace $2k-n$, we obtain \[ \operatorname{Gr}(k,n) \cong \{Q \in \operatorname{O}(n) : Q^{\mathsf{T}} = Q, \; \operatorname{tr}(Q) = 2k -n\}. \] As with the other two models, we show that differential geometric objects and operations -- tangent vector, metric, normal vector, exponential map, geodesic, parallel transport, gradient, Hessian, etc -- have closed-form analytic expressions that are computable with standard numerical linear algebra. In the proposed model, these expressions are considerably simpler, a result of representing $\operatorname{Gr}(k,n)$ as a linear section of a compact matrix Lie group $\operatorname{O}(n)$, and can be computed with at most one QR decomposition and one exponential of a special skew-symmetric matrix that takes only $O(nk(n-k))$ time. In particular, we completely avoid eigen- and singular value decompositions in our steepest descent, conjugate gradient, quasi-Newton, and Newton methods for the Grassmannian.

math.OC↗

arXiv2020

Symmetric Tensor Decompositions On Varieties

This paper discusses the problem of symmetric tensor decomposition on a given variety $X$: decomposing a symmetric tensor into the sum of tensor powers of vectors contained in $X$. In this paper, we first study geometric and algebraic properties of such decomposable tensors, which are crucial to the practical computations of such decompositions. For a given tensor, we also develop a criterion for the existence of a symmetric decomposition on $X$. Secondly and most importantly, we propose a method for computing symmetric tensor decompositions on an arbitrary $X$. As a specific application, Vandermonde decompositions for nonsymmetric tensors can be computed by the proposed algorithm.

math.NA↗

arXiv2019

Linear Convergence of an Alternating Polar Decomposition Method for Low Rank Orthogonal Tensor Approximations

Low rank orthogonal tensor approximation (LROTA) is an important problem in tensor computations and their applications. A classical and widely used algorithm is the alternating polar decomposition method (APD). In this article, an improved version iAPD of the classical APD is proposed. For the first time, all the following four fundamental properties are established for iAPD: (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual $O(1/k)$ for first order methods in optimization; (iii) more importantly, it converges $R$-linearly for a generic tensor without any assumption; (iv) for almost all LROTA problems, iAPD reduces to APD after finitely many iterations if it converges to a local minimizer.

math.OC↗