On GROUSE and Incremental SVD

GROUSE (Grassmannian Rank-One Update Subspace Estimation) is an incremental algorithm for identifying a subspace of Rn from a sequence of vectors in this subspace, where only a subset of components of each vector is revealed at each iteration. Recent analysis has shown that GROUSE converges locally at an expected linear rate, under certain assumptions. GROUSE has a similar flavor to the incremental singular value decomposition algorithm, which updates the SVD of a matrix following addition of a single column. In this paper, we modify the incremental SVD approach to handle missing data, and demonstrate that this modified approach is equivalent to GROUSE, for a certain choice of an algorithmic parameter

Relative Errors for Deterministic Low-Rank Matrix Approximations

We consider processing an n x d matrix A in a stream with row-wise updates according to a recent algorithm called Frequent Directions (Liberty, KDD 2013). This algorithm maintains an l x d matrix Q deterministically, processing each row in O(d l^2) time; the processing time can be decreased to O(d l) with a slight modification in the algorithm and a constant increase in space. We show that if one sets l = k+ k/eps and returns Q_k, a k x d matrix that is the best rank k approximation to Q, then we achieve the following properties: ||A - A_k||_F^2 <= ||A||_F^2 - ||Q_k||_F^2 <= (1+eps) ||A - A_k||_F^2 and where pi_{Q_k}(A) is the projection of A onto the rowspace of Q_k then ||A - pi_{Q_k}(A)||_F^2 <= (1+eps) ||A - A_k||_F^2. We also show that Frequent Directions cannot be adapted to a sparse version in an obvious way that retains the l original rows of the matrix, as opposed to a linear combination or sketch of the rows.Comment: 16 pages, 0 figure

Direct kernel biased discriminant analysis: a new content-based image retrieval relevance feedback algorithm

In recent years, a variety of relevance feedback (RF) schemes have been developed to improve the performance of content-based image retrieval (CBIR). Given user feedback information, the key to a RF scheme is how to select a subset of image features to construct a suitable dissimilarity measure. Among various RF schemes, biased discriminant analysis (BDA) based RF is one of the most promising. It is based on the observation that all positive samples are alike, while in general each negative sample is negative in its own way. However, to use BDA, the small sample size (SSS) problem is a big challenge, as users tend to give a small number of feedback samples. To explore solutions to this issue, this paper proposes a direct kernel BDA (DKBDA), which is less sensitive to SSS. An incremental DKBDA (IDKBDA) is also developed to speed up the analysis. Experimental results are reported on a real-world image collection to demonstrate that the proposed methods outperform the traditional kernel BDA (KBDA) and the support vector machine (SVM) based RF algorithms

Semantic Retrieval and Automatic Annotation: Linear Transformations, Correlation and Semantic Spaces

This paper proposes a new technique for auto-annotation and semantic retrieval based upon the idea of linearly mapping an image feature space to a keyword space. The new technique is compared to several related techniques, and a number of salient points about each of the techniques are discussed and contrasted. The paper also discusses how these techniques might actually scale to a real-world retrieval problem, and demonstrates this though a case study of a semantic retrieval technique being used on a real-world data-set (with a mix of annotated and unannotated images) from a picture library