Querying a Matrix Through Matrix-Vector Products

We consider algorithms with access to an unknown matrix M in F^{n x d} via matrix-vector products, namely, the algorithm chooses vectors v^1, ..., v^q, and observes Mv^1, ..., Mv^q. Here the v^i can be randomized as well as chosen adaptively as a function of Mv^1, ..., Mv^{i-1}. Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, this fundamental model does not appear to have been studied on its own, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas

Exact and efficient top-K inference for multi-target prediction by querying separable linear relational models

Many complex multi-target prediction problems that concern large target spaces are characterised by a need for efficient prediction strategies that avoid the computation of predictions for all targets explicitly. Examples of such problems emerge in several subfields of machine learning, such as collaborative filtering, multi-label classification, dyadic prediction and biological network inference. In this article we analyse efficient and exact algorithms for computing the top-$K$ predictions in the above problem settings, using a general class of models that we refer to as separable linear relational models. We show how to use those inference algorithms, which are modifications of well-known information retrieval methods, in a variety of machine learning settings. Furthermore, we study the possibility of scoring items incompletely, while still retaining an exact top-K retrieval. Experimental results in several application domains reveal that the so-called threshold algorithm is very scalable, performing often many orders of magnitude more efficiently than the naive approach

The Query Complexity of Mastermind with l_p Distances

Consider a variant of the Mastermind game in which queries are l_p distances, rather than the usual Hamming distance. That is, a codemaker chooses a hidden vector y in {-k,-k+1,...,k-1,k}^n and answers to queries of the form ||y-x||_p where x in {-k,-k+1,...,k-1,k}^n. The goal is to minimize the number of queries made in order to correctly guess y. In this work, we show an upper bound of O(min{n,(n log k)/(log n)}) queries for any real 10. Thus, essentially any approximation of this problem is as hard as finding the hidden vector exactly, up to constant factors. Finally, we show that for the noisy version of the problem, i.e., the setting when the codemaker answers queries with any q = (1 +/- epsilon)||y-x||_p, there is no query efficient algorithm

FLASH: Randomized Algorithms Accelerated over CPU-GPU for Ultra-High Dimensional Similarity Search

We present FLASH (\textbf{F}ast \textbf{L}SH \textbf{A}lgorithm for \textbf{S}imilarity search accelerated with \textbf{H}PC), a similarity search system for ultra-high dimensional datasets on a single machine, that does not require similarity computations and is tailored for high-performance computing platforms. By leveraging a LSH style randomized indexing procedure and combining it with several principled techniques, such as reservoir sampling, recent advances in one-pass minwise hashing, and count based estimations, we reduce the computational and parallelization costs of similarity search, while retaining sound theoretical guarantees. We evaluate FLASH on several real, high-dimensional datasets from different domains, including text, malicious URL, click-through prediction, social networks, etc. Our experiments shed new light on the difficulties associated with datasets having several million dimensions. Current state-of-the-art implementations either fail on the presented scale or are orders of magnitude slower than FLASH. FLASH is capable of computing an approximate k-NN graph, from scratch, over the full webspam dataset (1.3 billion nonzeros) in less than 10 seconds. Computing a full k-NN graph in less than 10 seconds on the webspam dataset, using brute-force ($n^2D$), will require at least 20 teraflops. We provide CPU and GPU implementations of FLASH for replicability of our results

To Index or Not to Index: Optimizing Exact Maximum Inner Product Search

Exact Maximum Inner Product Search (MIPS) is an important task that is widely pertinent to recommender systems and high-dimensional similarity search. The brute-force approach to solving exact MIPS is computationally expensive, thus spurring recent development of novel indexes and pruning techniques for this task. In this paper, we show that a hardware-efficient brute-force approach, blocked matrix multiply (BMM), can outperform the state-of-the-art MIPS solvers by over an order of magnitude, for some -- but not all -- inputs. In this paper, we also present a novel MIPS solution, MAXIMUS, that takes advantage of hardware efficiency and pruning of the search space. Like BMM, MAXIMUS is faster than other solvers by up to an order of magnitude, but again only for some inputs. Since no single solution offers the best runtime performance for all inputs, we introduce a new data-dependent optimizer, OPTIMUS, that selects online with minimal overhead the best MIPS solver for a given input. Together, OPTIMUS and MAXIMUS outperform state-of-the-art MIPS solvers by 3.2$\times$ on average, and up to 10.9$\times$, on widely studied MIPS datasets.Comment: 12 pages, 8 figures, 2 table