The streaming kk-mismatch problem

We consider the streaming complexity of a fundamental task in approximate pattern matching: the $k$-mismatch problem. It asks to compute Hamming distances between a pattern of length $n$ and all length-$n$ substrings of a text for which the Hamming distance does not exceed a given threshold $k$. In our problem formulation, we report not only the Hamming distance but also, on demand, the full \emph{mismatch information}, that is the list of mismatched pairs of symbols and their indices. The twin challenges of streaming pattern matching derive from the need both to achieve small working space and also to guarantee that every arriving input symbol is processed quickly. We present a streaming algorithm for the $k$-mismatch problem which uses $O(k\log{n}\log\frac{n}{k})$ bits of space and spends \ourcomplexity time on each symbol of the input stream, which consists of the pattern followed by the text. The running time almost matches the classic offline solution and the space usage is within a logarithmic factor of optimal. Our new algorithm therefore effectively resolves and also extends an open problem first posed in FOCS'09. En route to this solution, we also give a deterministic $O( k (\log \frac{n}{k} + \log |\Sigma|) )$-bit encoding of all the alignments with Hamming distance at most $k$ of a length-$n$ pattern within a text of length $O(n)$. This secondary result provides an optimal solution to a natural communication complexity problem which may be of independent interest.Comment: 27 page

Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case

Let $\mathbf{f}=(f\_1,\ldots,f\_m)$ and $\mathbf{g}=(g\_1,\ldots,g\_m)$ be two sets of $m\geq 1$ nonlinear polynomials over $\mathbb{K}[x\_1,\ldots,x\_n]$ ($\mathbb{K}$ being a field). We consider the computational problem of finding -- if any -- an invertible transformation on the variables mapping $\mathbf{f}$ to $\mathbf{g}$. The corresponding equivalence problem is known as {\tt Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental problem in multivariate cryptography. The main result is a randomized polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a particular case of importance in cryptography and somewhat justifying {\it a posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt IP1S} for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory, which involves to test the orthogonal simultaneous conjugacy of symmetric matrices. We show that we can essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to test the orthogonal simultaneous similarity of symmetric matrices; this latter problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding an invertible matrix in the linear space $\mathbb{K}^{n \times n}$ of $n \times n$ matrices over $\mathbb{K}$ and to compute the square root in a matrix algebra. While computing square roots of matrices can be done efficiently using numerical methods, it seems difficult to control the bit complexity of such methods. However, we present exact and polynomial-time algorithms for computing the square root in $\mathbb{K}^{n \times n}$ for various fields (including finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt IP1S} for quadratic instances. In particular, we provide a (complete) characterization of the automorphism group of homogeneous quadratic polynomials. Finally, we also consider the more general {\it Isomorphism of Polynomials} ({\tt IP}) problem where we allow an invertible linear transformation on the variables \emph{and} on the set of polynomials. A randomized polynomial-time algorithm for solving {\tt IP} when $\mathbf{f}=(x\_1^d,\ldots,x\_n^d)$ is presented. From an algorithmic point of view, the problem boils down to factoring the determinant of a linear matrix (\emph{i.e.}\ a matrix whose components are linear polynomials). This extends to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.Comment: Published in Journal of Complexity, Elsevier, 2015, pp.3

Tensor Analysis and Fusion of Multimodal Brain Images

Current high-throughput data acquisition technologies probe dynamical systems with different imaging modalities, generating massive data sets at different spatial and temporal resolutions posing challenging problems in multimodal data fusion. A case in point is the attempt to parse out the brain structures and networks that underpin human cognitive processes by analysis of different neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the multimodal, multi-scale nature of neuroimaging data is well reflected by a multi-way (tensor) structure where the underlying processes can be summarized by a relatively small number of components or "atoms". We introduce Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network notation in order to analyze these models. These diagrams not only clarify matrix and tensor EEG and fMRI time/frequency analysis and inverse problems, but also help understand multimodal fusion via Multiway Partial Least Squares and Coupled Matrix-Tensor Factorization. We show here, for the first time, that Granger causal analysis of brain networks is a tensor regression problem, thus allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI recordings shows the potential of the methods and suggests their use in other scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE