Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

The community's work to explain these observations has roughly fallen into either "biased data" or

Neural Information Processing Systems http://nips.cc/

We present quasicyclic principal component analysis (QPCA), a generalization of principal component analysis (PCA), that determines an optimized basis for a dataset in terms of families of shift-orthogonal principal vectors. This is of particular interest when analyzing cyclostationary data, whose cyclic structure is not exploited by the standard PCA algorithm. We first formulate QPCA as an optimization problem, which we show may be decomposed into a series of PCA problems in the frequency domain. We then formalize our solution as an explicit algorithm and analyze its computational complexity. Finally, we provide some examples of applications of QPCA to cyclostationary signal processing data, including an investigation of carrier pulse recovery, a presentation of methods for estimating an unknown oversampling rate, and a discussion of an appropriate approach for pre-processing data with a non-integer oversampling rate in order to better apply the QPCA algorithm.

We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. Such components can be computed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but this greedy procedure is suboptimal. We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to 1 from the optimal. Our algorithm is combinatorial and computes the desired components by solving multiple instances of the bipartite maximum weight matching problem. Its complexity grows as a low order polynomial in the ambient dimension of the input data, but exponentially in its rank. However, it can be effectively applied on a low-dimensional sketch of the input data. We evaluate our algorithm on real datasets and empirically demonstrate that in many cases it outperforms existing, deflation-based approaches.

Principal Component Analysis (PCA) is the workhorse tool for dimensionality reduction in this era of big data. While often overlooked, the purpose of PCA is not only to reduce data dimensionality, but also to yield features that are uncorrelated. This paper focuses on this dual objective of PCA, namely, dimensionality reduction and decorrelation of features, which requires estimating the eigenvectors of a data covariance matrix, as opposed to only estimating the subspace spanned by the eigenvectors. The ever-increasing volume of data in the modern world often requires storage of data samples across multiple machines, which precludes the use of centralized PCA algorithms. Although a few distributed solutions to the PCA problem have been proposed recently, convergence guarantees and/or communications overhead of these solutions remain a concern. With an eye towards communications efficiency, this paper introduces a feedforward neural network-based one time-scale distributed PCA algorithm termed Distributed Sanger's Algorithm (DSA) that estimates the eigenvectors of a data covariance matrix when data are distributed across an undirected and arbitrarily connected network of machines. Furthermore, the proposed algorithm is shown to converge linearly to a neighborhood of the true solution. Numerical results are also shown to demonstrate the efficacy of the proposed solution.

This paper considers the problem of estimating the principal eigenvector of a covariance matrix from independent and identically distributed data samples in streaming settings. The streaming rate of data in many contemporary applications can be high enough that a single processor cannot finish an iteration of existing methods for eigenvector estimation before a new sample arrives. This paper formulates and analyzes a distributed variant of the classical Krasulina's method (D-Krasulina) that can keep up with the high streaming rate of data by distributing the computational load across multiple processing nodes. The analysis shows that---under appropriate conditions---D-Krasulina converges to the principal eigenvector in an order-wise optimal manner; i.e., after receiving $M$ samples across all nodes, its estimation error can be $O(1/M)$. In order to reduce the network communication overhead, the paper also develops and analyzes a mini-batch extension of D-Krasulina, which is termed DM-Krasulina. The analysis of DM-Krasulina shows that it can also achieve order-optimal estimation error rates under appropriate conditions, even when some samples have to be discarded within the network due to communication latency. Finally, experiments are performed over synthetic and real-world data to validate the convergence behaviors of D-Krasulina and DM-Krasulina in high-rate streaming settings.

Principal component analysis is one of the most extensively studied methods for constructing linear low-dimensional representation of high-dimensional data. Modern applications such as privacy presevering distributedcomputations (Hardt and Roth (2013)), covariance estimtion of high-frequency data (Chang et al. (2018),Aït-Sahalia et al. (2010)), detecting power grid attacks (Bienstock and Shukla (2018), Escobar et al. (2018)) etc. require design of sub-linear time algorithms with low storage overhead. Existing workon PCA has focused on design and analysis of single-pass (streaming) algorithms with nearoptimal memoryand storage complexity assuming stationarity of the underlying data-generating process. However, physical systems generating data for such applications undergo rapid evolution. For example, dynamic market behaviour leads to time-series data with volatile covariance matrices. Our understanding of such physical system crucially relies on accurate estimation of the data generating space.

Stochastic optimization naturally arises in machine learning. Efficient algorithms with provable guarantees, however, are still largely missing, when the objective function is nonconvex and the data points are dependent. This paper studies this fundamental challenge through a streaming PCA problem for stationary time series data. Specifically, our goal is to estimate the principle component of time series data with respect to the covariance matrix of the stationary distribution. Computationally, we propose a variant of Oja's algorithm combined with downsampling to control the bias of the stochastic gradient caused by the data dependency. Theoretically, we quantify the uncertainty of our proposed stochastic algorithm based on diffusion approximations. This allows us to prove the asymptotic rate of convergence and further implies near optimal asymptotic sample complexity. Numerical experiments are provided to support our analysis.