最优凸提升稀疏相位恢复与基于原子矩阵范数正则化的主成分分析

Isometric Autoencoders

High dimensional data is often assumed to be concentrated on or near a low-dimensional manifold. Autoencoders (AE) is a popular technique to learn representations of such data by pushing it through a neural network with a low dimension bottleneck while minimizing a reconstruction error. Using high capacity AE often leads to a large collection of minimizers, many of which represent a low dimensional manifold that fits the data well but generalizes poorly. Two sources of bad generalization are: extrinsic, where the learned manifold possesses extraneous parts that are far from the data; and intri…

通过隐式正则化的免调参在线鲁棒主成分分析

Tuning-Free Online Robust Principal Component Analysis through Implicit Regularization

针对标准在线鲁棒主成分分析（OR-PCA）依赖数据集敏感调参的问题，提出利用修正梯度下降的隐式正则化效应实现免调参，在模拟与真实数据集上表现优于或等同于调参后的OR-PCA，提升了大规模数据集的可扩展性。

Consistent Estimation for PCA and Sparse Regression with Oblivious Outliers

We develop machinery to design efficiently computable and consistent estimators, achieving estimation error approaching zero as the number of observations grows, when facing an oblivious adversary that may corrupt responses in all but an $α$ fraction of the samples. As concrete examples, we investigate two problems: sparse regression and principal component analysis (PCA). For sparse regression, we achieve consistency for optimal sample size $n\gtrsim (k\log d)/α^2$ and optimal error rate $O(\sqrt{(k\log d)/(n\cdot α^2)})$ where $n$ is the number of observations, $d$ is the number of dimension…

Bandit Principal Component Analysis

We consider a partial-feedback variant of the well-studied online PCA problem where a learner attempts to predict a sequence of $d$-dimensional vectors in terms of a quadratic loss, while only having limited feedback about the environment's choices. We focus on a natural notion of bandit feedback where the learner only observes the loss associated with its own prediction. Based on the classical observation that this decision-making problem can be lifted to the space of density matrices, we propose an algorithm that is shown to achieve a regret of $O(d^{3/2}\sqrt{T})$ after $T$ rounds in the wo…