Aleksandr Beknazaryan
We show that neural networks with absolute value activation function and with the path norm, the depth, the width and the network weights having logarithmic dependence on $1/\varepsilon$ can $\varepsilon$-approximate functions that are analytic on certain regions of $\mathbb{C}^d$.
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We investigate the use of regularized Newton methods with adaptive norms for optimizing neural networks. This approach can be seen as a second-order counterpart of adaptive gradient methods, which we here show to be interpretable as first-order trust region methods with ellipsoidal constraints. In particular, we prove that the preconditioning matrix used in RMSProp and Adam satisfies the necessary conditions for provable convergence of second-order trust region methods with standard worst-case complexities on general non-convex objectives. Furthermore, we run experiments across different neura…
Recently, self-normalizing neural networks (SNNs) have been proposed with the intention to avoid batch or weight normalization. The key step in SNNs is to properly scale the exponential linear unit (referred to as SELU) to inherently incorporate normalization based on central limit theory. SELU is a monotonically increasing function, where it has an approximately constant negative output for large negative input. In this work, we propose a new activation function to break the monotonicity property of SELU while still preserving the self-normalizing property. Differently from SELU, the new func…
While modern convolutional neural networks achieve outstanding accuracy on many image classification tasks, they are, compared to humans, much more sensitive to image degradation. Here, we describe a variant of Batch Normalization, LocalNorm, that regularizes the normalization layer in the spirit of Dropout while dynamically adapting to the local image intensity and contrast at test-time. We show that the resulting deep neural networks are much more resistant to noise-induced image degradation, improving accuracy by up to three times, while achieving the same or slightly better accuracy on non…
Disagreement-Regularized Importance Sampling for Adversarial Label Corruption
本文提出DR-IS方法，通过损失排名分歧来应对标签污染问题，证明了样本集中污染率的控制，并在实验中验证了其在高norm攻击下的鲁棒性。
Learning the "blocking" structure is a central challenge for high dimensional data (e.g., gene expression data). Recently, a sparse singular value decomposition (SVD) has been used as a biclustering tool to achieve this goal. However, this model ignores the structural information between variables (e.g., gene interaction graph). Although typical graph-regularized norm can incorporate such prior graph information to get accurate discovery and better interpretability, it fails to consider the opposite effect of variables with different signs. Motivated by the development of sparse coding and gra…
A recently introduced technique for a sparse optimization problem called "safe screening" allows us to identify irrelevant variables in the early stage of optimization. In this paper, we first propose a flexible framework for safe screening based on the Fenchel-Rockafellar duality and then derive a strong safe screening rule for norm-regularized least squares by the framework. We call the proposed screening rule for norm-regularized least squares "dynamic Sasvi" because it can be interpreted as a generalization of Sasvi. Unlike the original Sasvi, it does not require the exact solution of a mo…
Kernel logistic regression (KLR) is a powerful classification method widely applied across diverse domains. In many real-world scenarios, indefinite kernels capture more domain-specific structural information than positive definite kernels. This paper proposes a novel $L_1$-norm regularized indefinite kernel logistic regression (RIKLR) model, which extends the existing IKLR framework by introducing sparsity via an $L_1$-norm penalty. The introduction of this regularization enhances interpretability and generalization while introducing nonsmoothness and nonconvexity into the optimization landsc…
Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion
本文针对核范数正则化矩阵补全，在目标矩阵顶部特征空间不连贯假设下，首次推导出恢复满秩矩阵最佳低秩逼近的相对误差上界，弥补了现有加性误差界无法刻画完美恢复及特征值偏斜分布的不足。
This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian for the loss function, we derive an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, under the noisy and full sample setting we establish its KL property of exponent $1/2$ on its global minimizer set, and under the noisy and partial sample s…
A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the l1 norm can be applied to improve the learning rate estimate of l1-regularization in machine learning.
This paper is concerned with the factorization form of the rank regularized loss minimization problem. To cater for the scenario in which only a coarse estimation is available for the rank of the true matrix, an $\ell_{2,0}$-norm regularized term is added to the factored loss function to reduce the rank adaptively; and account for the ambiguities in the factorization, a balanced term is then introduced. For the least squares loss, under a restricted condition number assumption on the sampling operator, we establish the KL property of exponent $1/2$ of the nonsmooth factored composite function …
Implicitly Normalized Online PCA: A Regularized Algorithm with Exact High-Dimensional Dynamics
隐式归一化在线主成分分析（INO-PCA）通过动态范数演变和正则化更新，在高维极限下实现对问题相关信息的有效编码和利用，优于Oja算法并适应非平稳环境。
Subspace segmentation assumes that data comes from the union of different subspaces and the purpose of segmentation is to partition the data into the corresponding subspace. Low-rank representation (LRR) is a classic spectral-type method for solving subspace segmentation problems, that is, one first obtains an affinity matrix by solving a LRR model and then performs spectral clustering for segmentation. This paper proposes a group norm regularized factorization model (GNRFM) inspired by the LRR model for subspace segmentation and then designs an Accelerated Augmented Lagrangian Method (AALM) a…
This work proposes an adaptive trace lasso regularized L1-norm based graph cut method for dimensionality reduction of Hyperspectral images, called as `Trace Lasso-L1 Graph Cut' (TL-L1GC). The underlying idea of this method is to generate the optimal projection matrix by considering both the sparsity as well as the correlation of the data samples. The conventional L2-norm used in the objective function is sensitive to noise and outliers. Therefore, in this work L1-norm is utilized as a robust alternative to L2-norm. Besides, for further improvement of the results, we use a penalty function of t…
This paper is concerned with the column $\ell_{2,0}$-regularized factorization model of low-rank matrix recovery problems and its computation. The column $\ell_{2,0}$-norm of factor matrices is introduced to promote column sparsity of factors and low-rank solutions. For this nonconvex discontinuous optimization problem, we develop an alternating majorization-minimization (AMM) method with extrapolation, and a hybrid AMM in which a majorized alternating proximal method is proposed to seek an initial factor pair with less nonzero columns and the AMM with extrapolation is then employed to minimiz…
Nuclear Norm-Regularized Bayesian Matrix Completion
针对核范数正则化贝叶斯矩阵补全，提出首个具有非渐近保证的多项式时间采样器，通过离散化噪声精度并利用热力学积分构建后验，证明了可行性。
Learning rates for least-squares regression are typically expressed in terms of $L_2$-norms. In this paper we extend these rates to norms stronger than the $L_2$-norm without requiring the regression function to be contained in the hypothesis space. In the special case of Sobolev reproducing kernel Hilbert spaces used as hypotheses spaces, these stronger norms coincide with fractional Sobolev norms between the used Sobolev space and $L_2$. As a consequence, not only the target function but also some of its derivatives can be estimated without changing the algorithm. From a technical point of v…
Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm
本文研究无穷维向量值核岭回归，利用向量值插值空间的张量积构造刻画光滑性，在错误设定下建立与输出维数无关且多数情形最优的连续范数速率。
Optimal a priori estimates are derived for the population risk, also known as the generalization error, of a regularized residual network model. An important part of the regularized model is the usage of a new path norm, called the weighted path norm, as the regularization term. The weighted path norm treats the skip connections and the nonlinearities differently so that paths with more nonlinearities are regularized by larger weights. The error estimates are a priori in the sense that the estimates depend only on the target function, not on the parameters obtained in the training process. The…
l1-norm regularized l1-norm best-fit lines
针对NP难的稀疏鲁棒一维子空间估计问题，提出基于线性松弛与比值排序的l1范数正则化l1范数拟合算法，具备多项式复杂度、无需初始化、可复现、速度快、稀疏性与拟合权衡更优的特点。
Sparse learning has recently received increasing attention in many areas including machine learning, statistics, and applied mathematics. The mixed-norm regularization based on the l1q norm with q>1 is attractive in many applications of regression and classification in that it facilitates group sparsity in the model. The resulting optimization problem is, however, challenging to solve due to the inherent structure of the mixed-norm regularization. Existing work deals with special cases with q=1, 2, infinity, and they cannot be easily extended to the general case. In this paper, we propose an e…
This article develops a general theory for minimum norm interpolating estimators and regularized empirical risk minimizers (RERM) in linear models in the presence of additive, potentially adversarial, errors. In particular, no conditions on the errors are imposed. A quantitative bound for the prediction error is given, relating it to the Rademacher complexity of the covariates, the norm of the minimum norm interpolator of the errors and the size of the subdifferential around the true parameter. The general theory is illustrated for Gaussian features and several norms: The $\ell_1$, $\ell_2$, g…
This paper explores the recently proposed Graph Convolutional Network architecture proposed in (Kipf & Welling, 2016) The key points of their work is summarized and their results are reproduced. Graph regularization and alternative graph convolution approaches are explored. I find that explicit graph regularization was correctly rejected by (Kipf & Welling, 2016). I attempt to improve the performance of GCN by approximating a k-step transition matrix in place of the normalized graph laplacian, but I fail to find positive results. Nonetheless, the performance of several configurations of this G…
Schatten-$p$ Quasi-Norm Regularized Matrix Optimization via Iterative Reweighted Singular Value Minimization
本文研究了Schatten-p准范数正则化的矩阵最小化问题，提出了一种迭代加权奇异值最小化方法，证明了其收敛性并展示了其在解质量和速度上的优势。
Nuclear Norm Regularized Estimation of Panel Regression Models
本文提出基于核范数正则化的面板回归模型估计方法，解决了交互固定效应模型的识别问题，并通过迭代步骤实现与传统最小二乘估计的渐近等价。
Prior knowledge on properties of a target model often come as discrete or combinatorial descriptions. This work provides a unified computational framework for defining norms that promote such structures. More specifically, we develop associated tools for optimization involving such norms given only the orthogonal projection oracle onto the non-convex set of desired models. As an example, we study a norm, which we term the doubly-sparse norm, for promoting vectors with few nonzero entries taking only a few distinct values. We further discuss how the K-means algorithm can serve as the underlying…
Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone $K$, a norm $\|\cdot\|$ and a smooth convex function $f$, we want either 1) to minimize the norm over the intersection of the cone and a level set of $f$, or 2) to minimize over the cone the sum of $f$ and a multiple of the norm. We focus on the case where (a) the dimension of the problem is too large to allow for interior point algorithms, (b) $\|\cdot\|$ is "too complicated" to allow for computationally cheap Bregman projections required in the first-orde…
We study the estimation of $β$ for the nonlinear model $y = f(X\sp{\top}β) + ε$ when $f$ is a nonlinear transformation that is known, $β$ has sparse nonzero coordinates, and the number of observations can be much smaller than that of parameters ($n\ll p$). We show that in order to bound the $L_2$ error of the $L_0$ regularized estimator $\hatβ$, i.e., $\|\hatβ- β\|_2$, it is sufficient to establish two conditions. Based on this, we obtain bounds of the $L_2$ error for (1) $L_0$ regularized maximum likelihood estimation (MLE) for exponential linear models and (2) $L_0$ regularized least square …
Dynamic Proximal Gradient Algorithms for Schatten-$p$ Quasi-Norm Regularized Problems
本文提出一种动态近端梯度算法，通过Cayley变换降低计算复杂性，并在不同p值下证明了算法的亚线性或线性收敛性，同时验证了其高效性。
In this work we study input gradient regularization of deep neural networks, and demonstrate that such regularization leads to generalization proofs and improved adversarial robustness. The proof of generalization does not overcome the curse of dimensionality, but it is independent of the number of layers in the networks. The adversarial robustness regularization combines adversarial training, which we show to be equivalent to Total Variation regularization, with Lipschitz regularization. We demonstrate empirically that the regularized models are more robust, and that gradient norms of images …
A MAP approach for $\ell_q$-norm regularized sparse parameter estimation using the EM algorithm
本文基于EM算法重新考虑了通过最大后验（MAP）准则进行贝叶斯参数估计的问题，利用合适先验分布引入稀疏性促进惩罚项，通过方差-均值高斯混合（VMGM）描述先验分布，解决非线性和隐变量优化问题，并通过仿真比较了坐标下降算法的性能。
Hankel Matrix Nuclear Norm Regularized Tensor Completion for $N$-dimensional Exponential Signals
本文提出了一种基于Hankel矩阵核范数正则化的张量补全方法，用于恢复N维（特别是N≥3）的指数信号，通过同时利用CANDECOMP/PARAFAC结构和因子向量的指数结构，有效恢复有限样本下的完整信号。
Given a limited number of entries from the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, recovery of the low-rank and sparse components is a fundamental task subsuming compressed sensing, matrix completion, and principal components pursuit. This paper develops algorithms for distributed sparsity-regularized rank minimization over networks, when the nuclear- and $\ell_1$-norm are used as surrogates to the rank and nonzero entry counts of the sought matrices, respectively. While nuclear-norm minimization has well-documented merits wh…
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