Low-rank approximations are popular methods to reduce the high computational cost of algorithms involving large-scale kernel matrices. The success of low-rank methods hinges on the matrix rank, and in practice, these methods are effective even for high-dimensional datasets. The practical success has elicited the theoretical analysis of the function rank in this paper, which is an upper bound of...

A new submodule clustering method via sparse and lowrank representation for multi-way data is proposed in this paper. Instead of reshaping multi-way data into vectors, this method maintains their natural orders to preserve data intrinsic structures, e.g., image data kept as matrices. To implement clustering, the multi-way data, viewed as tensors, are represented by the proposed tensor sparse an...

This paper studies the problem of learning human action attributes based on union-of-subspaces model. It puts forth an extension of the low-rank representation (LRR) model, termed the hierarchical clustering-aware structure-constrained low-rank representation (HCSLRR) model, for unsupervised learning of human action attributes from video data. The effectiveness of the proposed model is demonstr...

Sentences are important semantic units of natural language. A generic, distributional representation of sentences that can capture the latent semantics is beneficial to multiple downstream applications. We observe a simple geometry of sentences – the word representations of a given sentence (on average 10.23 words in all SemEval datasets with a standard deviation 4.84) roughly lie in a low-rank...

We present a multi-scale version of low rank matrix decomposition. Our motivation comes from imaging applications, in which image sequences are correlated locally on several scales in space and time rather than globally. We model our data matrix as a sum of matrices, where each matrix has increasing scales of locally low-rank matrices. Using this multi-scale modeling, we can capture different s...

Let M be a tropical matrix (k + x) × (k + x ′) for some k, x , x ′ ∈ N − {0} with tropical rank k. We show that Kapranov rank is k too if x and x ′ are not too big; namely if we are in one of the following cases: a) k ≥ 6 and x , x ′ ≤ 2 b) k = 4, 5, x ≤ 2 and x ′ ≤ 3 (or obviuosly the converse, that is x ≤ 3 and x ′ ≤ 2) c) k = 3 and either x , x ′ ≤ 3 or x ≤ 2 and x ′ ≤ 4 (or obviuosly the co...