Ziyan Luo

Liping Zhang, Liqun Qi, Ziyan Luo

It is well known that the dominant eigenvalue of a real essentially nonnegative matrix is a convex function of its diagonal entries. This convexity is of practical importance in population biology, graph theory, demography, analytic hierarchy process and so on. In this paper, the concept of essentially nonnegativity is extended from matrices to higher order tensors, and the convexity and log convexity of dominant eigenvalues for such a class of tensors are established. Particularly, for any nonnegative tensor, the spectral radius turns out to be the dominant eigenvalue and hence possesses these convexities. Finally, an algorithm is given to calculate the dominant eigenvalue, and numerical results are reported to show the effectiveness of the proposed algorithm.

Wang Chen, Ziyan Luo, Shuangyue Wang

Input features are conventionally represented as vectors, matrices, or third order tensors in the real field, for color image classification. Inspired by the success of quaternion data modeling for color images in image recovery and denoising tasks, we propose a novel classification method for color image classification, named as the Low-rank Support Quaternion Matrix Machine (LSQMM), in which the RGB channels are treated as pure quaternions to effectively preserve the intrinsic coupling relationships among channels via the quaternion algebra. For the purpose of promoting low-rank structures resulting from strongly correlated color channels, a quaternion nuclear norm regularization term, serving as a natural extension of the conventional matrix nuclear norm to the quaternion domain, is added to the hinge loss in our LSQMM model. An Alternating Direction Method of Multipliers (ADMM)-based iterative algorithm is designed to effectively resolve the proposed quaternion optimization model. Experimental results on multiple color image classification datasets demonstrate that our proposed classification approach exhibits advantages in classification accuracy, robustness and computational efficiency, compared to several state-of-the-art methods using support vector machines, support matrix machines, and support tensor machines.

Ziyan Luo, Tianwei Ni, Pierre-Luc Bacon et al.

A key approach to state abstraction is approximating behavioral metrics (notably, bisimulation metrics) in the observation space and embedding these learned distances in the representation space. While promising for robustness to task-irrelevant noise, as shown in prior work, accurately estimating these metrics remains challenging, requiring various design choices that create gaps between theory and practice. Prior evaluations focus mainly on final returns, leaving the quality of learned metrics and the source of performance gains unclear. To systematically assess how metric learning works in deep reinforcement learning (RL), we evaluate five recent approaches, unified conceptually as isometric embeddings with varying design choices. We benchmark them with baselines across 20 state-based and 14 pixel-based tasks, spanning 370 task configurations with diverse noise settings. Beyond final returns, we introduce the evaluation of a denoising factor to quantify the encoder's ability to filter distractions. To further isolate the effect of metric learning, we propose and evaluate an isolated metric estimation setting, in which the encoder is influenced solely by the metric loss. Finally, we release an open-source, modular codebase to improve reproducibility and support future research on metric learning in deep RL.

Martin Klissarov, Akhil Bagaria, Ziyan Luo et al.

Developing agents capable of exploring, planning and learning in complex open-ended environments is a grand challenge in artificial intelligence (AI). Hierarchical reinforcement learning (HRL) offers a promising solution to this challenge by discovering and exploiting the temporal structure within a stream of experience. The strong appeal of the HRL framework has led to a rich and diverse body of literature attempting to discover a useful structure. However, it is still not clear how one might define what constitutes good structure in the first place, or the kind of problems in which identifying it may be helpful. This work aims to identify the benefits of HRL from the perspective of the fundamental challenges in decision-making, as well as highlight its impact on the performance trade-offs of AI agents. Through these benefits, we then cover the families of methods that discover temporal structure in HRL, ranging from learning directly from online experience to offline datasets, to leveraging large language models (LLMs). Finally, we highlight the challenges of temporal structure discovery and the domains that are particularly well-suited for such endeavours.

Shenglong Zhou, Ouya Wang, Ziyan Luo et al.

The recent advancement of foundation models (FMs) has brought about a paradigm shift, revolutionizing various sectors worldwide. The popular optimizers used to train these models are stochastic gradient descent-based algorithms, which face inherent limitations, such as slow convergence and stringent assumptions for convergence. In particular, data heterogeneity arising from distributed settings poses significant challenges to their theoretical and numerical performance. This paper develops an algorithm, PISA (Preconditioned Inexact Stochastic Alternating Direction Method of Multipliers). Grounded in rigorous theoretical guarantees, the algorithm converges under the sole assumption of Lipschitz continuity of the gradient on a bounded region, thereby removing the need for other conditions commonly imposed by stochastic methods. This capability enables the proposed algorithm to tackle the challenge of data heterogeneity effectively. Moreover, the algorithmic architecture enables scalable parallel computing and supports various preconditions, such as second-order information, second moment, and orthogonalized momentum by Newton-Schulz iterations. Incorporating the latter two preconditions in PISA yields two computationally efficient variants: SISA and NSISA. Comprehensive experimental evaluations for training or fine-tuning diverse deep models, including vision models, large language models, reinforcement learning models, generative adversarial networks, and recurrent neural networks, demonstrate superior numerical performance of SISA and NSISA compared to various state-of-the-art optimizers.

Ziyan Luo, Xujie Si

Solving Constrained Horn Clauses (CHCs) is a fundamental challenge behind a wide range of verification and analysis tasks. Data-driven approaches show great promise in improving CHC solving without the painstaking manual effort of creating and tuning various heuristics. However, a large performance gap exists between data-driven CHC solvers and symbolic reasoning-based solvers. In this work, we develop a simple but effective framework, "Chronosymbolic Learning", which unifies symbolic information and numerical data points to solve a CHC system efficiently. We also present a simple instance of Chronosymbolic Learning with a data-driven learner and a BMC-styled reasoner. Despite its relative simplicity, experimental results show the efficacy and robustness of our tool. It outperforms state-of-the-art CHC solvers on a dataset consisting of 288 benchmarks, including many instances with non-linear integer arithmetics.

Yunfang Fu, Qiuqi Ruan, Ziyan Luo et al.

In this paper, a novel approach via embedded tensor manifold regularization for 2D+3D facial expression recognition (FERETMR) is proposed. Firstly, 3D tensors are constructed from 2D face images and 3D face shape models to keep the structural information and correlations. To maintain the local structure (geometric information) of 3D tensor samples in the low-dimensional tensors space during the dimensionality reduction, the $\ell_0$-norm of the core tensors and a tensor manifold regularization scheme embedded on core tensors are adopted via a low-rank truncated Tucker decomposition on the generated tensors. As a result, the obtained factor matrices will be used for facial expression classification prediction. To make the resulting tensor optimization more tractable, $\ell_1$-norm surrogate is employed to relax $\ell_0$-norm and hence the resulting tensor optimization problem has a nonsmooth objective function due to the $\ell_1$-norm and orthogonal constraints from the orthogonal Tucker decomposition. To efficiently tackle this tensor optimization problem, we establish the first-order optimality condition in terms of stationary points, and then design a block coordinate descent (BCD) algorithm with convergence analysis and the computational complexity. Numerical results on BU-3DFE database and Bosphorus databases demonstrate the effectiveness of our proposed approach.

Shenglong Zhou, Naihua Xiu, Ziyan Luo et al.

This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices. We first benefit from a convex optimization which develops $l_1$-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property. For the proposed estimator, we then prove that with large probability, the Frobenious norm of the estimation rate can be of order $O(\sqrt{s(\log{r})/n})$ under a mild case, where $s$ and $r$ denote the number of sparse entries and the rank of the population covariance respectively, $n$ notes the sample capacity. Finally an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem, and meantime merits of the approach are also illustrated by practicing numerical simulations.