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Jianze Li

Publications and source records attributed to Jianze Li.

32 records · Page 2Linked to original sources

Unveiling and Mitigating Backdoor Vulnerabilities based on Unlearning Weight Changes and Backdoor Activeness

The security threat of backdoor attacks is a central concern for deep neural networks (DNNs). Recently, without poisoned data, unlearning models with clean data and then learning a pruning mask have contributed to backdoor defense. Additionally, vanilla fine-tuning with those clean data can help recover the lost clean accuracy. However, the behavior of clean unlearning is still under-explored, and vanilla fine-tuning unintentionally induces back the backdoor effect. In this work, we first investigate model unlearning from the perspective of weight changes and gradient norms, and find two interesting observations in the backdoored model: 1) the weight changes between poison and clean unlearning are positively correlated, making it possible for us to identify the backdoored-related neurons without using poisoned data; 2) the neurons of the backdoored model are more active (i.e., larger changes in gradient norm) than those in the clean model, suggesting the need to suppress the gradient norm during fine-tuning. Then, we propose an effective two-stage defense method. In the first stage, an efficient Neuron Weight Change (NWC)-based Backdoor Reinitialization is proposed based on observation 1). In the second stage, based on observation 2), we design an Activeness-Aware Fine-Tuning to replace the vanilla fine-tuning. Extensive experiments, involving eight backdoor attacks on three benchmark datasets, demonstrate the superior performance of our proposed method compared to recent state-of-the-art backdoor defense approaches.

cs.CR↗

Convergence of gradient-based block coordinate descent algorithms for non-orthogonal joint approximate diagonalization of matrices

In this paper, we propose a gradient-based block coordinate descent (BCD-G) framework to solve the joint approximate diagonalization of matrices defined on the product of the complex Stiefel manifold and the special linear group. Instead of the cyclic fashion, we choose a block optimization based on the Riemannian gradient. To update the first block variable in the complex Stiefel manifold, we use the well-known line search descent method. To update the second block variable in the special linear group, based on four kinds of different elementary transformations, we construct three classes: GLU, GQU and GU, and then get three BCD-G algorithms: BCD-GLU, BCD-GQU and BCD-GU. We establish the global and weak convergence of these three algorithms using the Łojasiewicz gradient inequality under the assumption that the iterates are bounded. We also propose a gradient-based Jacobi-type framework to solve the joint approximate diagonalization of matrices defined on the special linear group. As in the BCD-G case, using the GLU and GQU classes of elementary transformations, we focus on the Jacobi-GLU and Jacobi-GQU algorithms and establish their global and weak convergence. All the algorithms and convergence results described in this paper also apply to the real case.

math.NA↗

Jacobi-type algorithms for homogeneous polynomial optimization on Stiefel manifolds with applications to tensor approximations

This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result in [Usevich et al. SIOPT 2020], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multi-block Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms.

math.OC↗

TAOTF: A Two-stage Approximately Orthogonal Training Framework in Deep Neural Networks

The orthogonality constraints, including the hard and soft ones, have been used to normalize the weight matrices of Deep Neural Network (DNN) models, especially the Convolutional Neural Network (CNN) and Vision Transformer (ViT), to reduce model parameter redundancy and improve training stability. However, the robustness to noisy data of these models with constraints is not always satisfactory. In this work, we propose a novel two-stage approximately orthogonal training framework (TAOTF) to find a trade-off between the orthogonal solution space and the main task solution space to solve this problem in noisy data scenarios. In the first stage, we propose a novel algorithm called polar decomposition-based orthogonal initialization (PDOI) to find a good initialization for the orthogonal optimization. In the second stage, unlike other existing methods, we apply soft orthogonal constraints for all layers of DNN model. We evaluate the proposed model-agnostic framework both on the natural image and medical image datasets, which show that our method achieves stable and superior performances to existing methods.

cs.CV↗

Data-free Backdoor Removal based on Channel Lipschitzness

Recent studies have shown that Deep Neural Networks (DNNs) are vulnerable to the backdoor attacks, which leads to malicious behaviors of DNNs when specific triggers are attached to the input images. It was further demonstrated that the infected DNNs possess a collection of channels, which are more sensitive to the backdoor triggers compared with normal channels. Pruning these channels was then shown to be effective in mitigating the backdoor behaviors. To locate those channels, it is natural to consider their Lipschitzness, which measures their sensitivity against worst-case perturbations on the inputs. In this work, we introduce a novel concept called Channel Lipschitz Constant (CLC), which is defined as the Lipschitz constant of the mapping from the input images to the output of each channel. Then we provide empirical evidences to show the strong correlation between an Upper bound of the CLC (UCLC) and the trigger-activated change on the channel activation. Since UCLC can be directly calculated from the weight matrices, we can detect the potential backdoor channels in a data-free manner, and do simple pruning on the infected DNN to repair the model. The proposed Channel Lipschitzness based Pruning (CLP) method is super fast, simple, data-free and robust to the choice of the pruning threshold. Extensive experiments are conducted to evaluate the efficiency and effectiveness of CLP, which achieves state-of-the-art results among the mainstream defense methods even without any data. Source codes are available at https://github.com/rkteddy/channel-Lipschitzness-based-pruning.

cs.LG↗

Projectively and weakly simultaneously diagonalizable matrices and their applications

Characterizing simultaneously diagonalizable (SD) matrices has been receiving considerable attention in the recent decades due to its wide applications and its role in matrix analysis. However, the notion of SD matrices is arguably still restrictive for wider applications. In this paper, we consider two error measures related to the simultaneous diagonalization of matrices, and propose several new variants of SD thereof; in particular, TWSD, TWSD-B, T_{m,n}-SD (SDO), DWSD and D_{m,n}-SD (SDO). Those are all weaker forms of SD. We derive various sufficient and/or necessary conditions of them under different assumptions, and show the relationships between these new notions. Finally, we discuss the applications of these new notions in, e.g., quadratically constrained quadratic programming (QCQP) and independent component analysis (ICA).

math.NA↗

Jacobi-type algorithm for low rank orthogonal approximation of symmetric tensors and its convergence analysis

In this paper, we propose a Jacobi-type algorithm to solve the low rank orthogonal approximation problem of symmetric tensors. This algorithm includes as a special case the well-known Jacobi CoM2 algorithm for the approximate orthogonal diagonalization problem of symmetric tensors. We first prove the weak convergence of this algorithm, \textit{i.e.} any accumulation point is a stationary point. Then we study the global convergence of this algorithm under a gradient based ordering for a special case: the best rank-2 orthogonal approximation of 3rd order symmetric tensors, and prove that an accumulation point is the unique limit point under some conditions. Numerical experiments are presented to show the efficiency of this algorithm.

math.NA↗

On the convergence of Jacobi-type algorithms for Independent Component Analysis

Jacobi-type algorithms for simultaneous approximate diagonalization of real (or complex) symmetric tensors have been widely used in independent component analysis (ICA) because of their good performance. One natural way of choosing the index pairs in Jacobi-type algorithms is the classical cyclic ordering, while the other way is based on the Riemannian gradient in each iteration. In this paper, we mainly review in an accessible manner our recent results in a series of papers about weak and global convergence of these Jacobi-type algorithms. These results are mainly based on the Lojasiewicz gradient inequality.

math.NA↗

Polar decomposition based algorithms on the product of Stiefel manifolds with applications in tensor approximation

In this paper, we propose a general algorithmic framework to solve a class of optimization problems on the product of complex Stiefel manifolds based on the matrix polar decomposition. We establish the weak convergence, global convergence and linear convergence rate of this general algorithmic approach using the Łojasiewicz gradient inequality and the Morse-Bott property. This general algorithm and its convergence results are applied to the simultaneous approximate tensor diagonalization and simultaneous approximate tensor compression, which include as special cases the low rank orthogonal approximation, best rank-1 approximation and low multilinear rank approximation for higher order complex tensors. We also present a symmetric variant of this general algorithm to solve a symmetric variant of this class of optimization models, which essentially optimizes over a single Stiefel manifold. We establish its weak convergence, global convergence and linear convergence rate in a similar way. This symmetric variant and its convergence results are applied to the simultaneous approximate symmetric tensor diagonalization, which includes as special cases the low rank symmetric orthogonal approximation and best symmetric rank-1 approximation for higher order complex symmetric tensors. It turns out that well-known algorithms such as LROAT, S-LROAT, HOPM, S-HOPM are all special cases of this general algorithmic framework and its symmetric variant, and our convergence results subsume the results found in the literature designed for those special cases. All the algorithms and convergence results in this paper also apply to the real case.

math.NA↗

Point cloud denoising based on tensor Tucker decomposition

In this paper, we propose a new algorithm for point cloud denoising based on the tensor Tucker decomposition. We first represent the local surface patches of a noisy point cloud to be matrices by their distances to a reference point, and stack the similar patch matrices to be a 3rd order tensor. Then we use the Tucker decomposition to compress this patch tensor to be a core tensor of smaller size. We consider this core tensor as the frequency domain and remove the noise by manipulating the hard thresholding. Finally, all the fibers of the denoised patch tensor are placed back, and the average is taken if there are more than one estimators overlapped. The experimental evaluation shows that the proposed algorithm outperforms the state-of-the-art graph Laplacian regularized (GLR) algorithm when the Gaussian noise is high ($σ=0.1$), and the GLR algorithm is better in lower noise cases ($σ=0.04, 0.05, 0.08$).

cs.CV↗

On approximate diagonalization of third order symmetric tensors by orthogonal transformations

In this paper, we study the approximate orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to the stationary points of this problem. We study the relationships between these classes, and other well-known objects, such as tensor Z-eigenvalue and Z-eigenvector. We also prove results on convergence of the cyclic Jacobi (or Jacobi CoM2) algorithm.

math.NA↗

Globally convergent Jacobi-type algorithms for simultaneous orthogonal symmetric tensor diagonalization

In this paper, we consider a family of Jacobi-type algorithms for simultaneous orthogonal diagonalization problem of symmetric tensors. For the Jacobi-based algorithm of [SIAM J. Matrix Anal. Appl., 2(34):651--672, 2013], we prove its global convergence for simultaneous orthogonal diagonalization of symmetric matrices and 3rd-order tensors. We also propose a new Jacobi-based algorithm in the general setting and prove its global convergence for sufficiently smooth functions.

math.NA↗

A construction of inductive limit for operator system

In this paper, we show a construction of inductive limit for operator system based on Archimedeanization. This inductive limit may be not a closed operator system. We prove that many nuclearity properties could be preserved by a special case of this inductive limit.

math.OA↗