An asymptotic bound for secant varieties of Segre varieties

This paper studies the defectivity of secant varieties of Segre varieties. We prove that there exists an asymptotic lower estimate for the greater non-defective secant variety (without filling the ambient space) of any given Segre variety. In particular, we prove that the ratio between the greater non-defective secant variety of a Segre variety and its expected rank is lower bounded by a value depending just on the number of factors of the Segre variety. Moreover, in the final section, we present some results obtained by explicit computation, proving the non-defectivity of all the secant varieties of Segre varieties of the shape (P^n)^4, with 1 < n < 11, except at most \sigma_199((P^8)^4) and \sigma_357((P^10)^4).Comment: 14 page

Finding Exogenous Variables in Data with Many More Variables than Observations

Many statistical methods have been proposed to estimate causal models in classical situations with fewer variables than observations (p<n, p: the number of variables and n: the number of observations). However, modern datasets including gene expression data need high-dimensional causal modeling in challenging situations with orders of magnitude more variables than observations (p>>n). In this paper, we propose a method to find exogenous variables in a linear non-Gaussian causal model, which requires much smaller sample sizes than conventional methods and works even when p>>n. The key idea is to identify which variables are exogenous based on non-Gaussianity instead of estimating the entire structure of the model. Exogenous variables work as triggers that activate a causal chain in the model, and their identification leads to more efficient experimental designs and better understanding of the causal mechanism. We present experiments with artificial data and real-world gene expression data to evaluate the method.Comment: A revised version of this was published in Proc. ICANN201

A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the Orientation Distribution Function in Diffusion MRI

International audienceIn this paper we extract the geometric characteristics from an antipodally symmetric spherical function (ASSF), which can be de- scribed equivalently in the spherical harmonic (SH) basis, in the symmet- ric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. All three bases span the same vector space and are bijective when the rank of the SH series equals the order of the ST and equals the degree of the HP. We show, therefore, how it is possible to extract the maxima and minima of an ASSF by computing the stationary points of a constrained HP. In Diffusion MRI, the Orientation Distribution Function (ODF), repre- sents a state of the art reconstruction method whose maxima are aligned with the dominant fiber bundles. It is, therefore, important to be able to correctly estimate these maxima to detect the fiber directions. The ODF is an ASSF. To illustrate the potential of our method, we take up the example of the ODF, and extract its maxima to detect the fiber directions. Thanks to our method we are able to extract the maxima without limiting our search to a discrete set of values on the sphere, but by searching the maxima of a continuous function. Our method is also general, not dependent on the ODF, and the framework we present can be applied to any ASSF described in one of the three bases

A Canonical Genetic Algorithm for Blind Inversion of Linear Channels

It is well known the relationship between source separation and blind deconvolution: If a filtered version of an unknown i.i.d. signal is observed, temporal independence between samples can be used to retrieve the original signal, in the same manner as spatial independence is used for source separation. In this paper we propose the use of a Genetic Algorithm (GA) to blindly invert linear channels. The use of GA is justified in the case of small number of samples, where other gradient-like methods fails because of poor estimation of statistics

Forward Analysis and Model Checking for Trace Bounded WSTS

We investigate a subclass of well-structured transition systems (WSTS), the bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete deterministic ones, which we claim provide an adequate basis for the study of forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth. Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered previously for the termination of forward analysis, boundedness is decidable. Boundedness turns out to be a valuable restriction for WSTS verification, as we show that it further allows to decide all $\omega$-regular properties on the set of infinite traces of the system

Decomposition of homogeneous polynomials with low rank

Let $F$ be a homogeneous polynomial of degree $d$ in $m+1$ variables defined over an algebraically closed field of characteristic zero and suppose that $F$ belongs to the $s$-th secant varieties of the standard Veronese variety $X_{m,d}\subset \mathbb{P}^{{m+d\choose d}-1}$ but that its minimal decomposition as a sum of $d$-th powers of linear forms $M_1, ..., M_r$ is $F=M_1^d+... + M_r^d$ with $r>s$. We show that if $s+r\leq 2d+1$ then such a decomposition of $F$ can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of $F$ if the rank is at most $d$ and a mild condition is satisfied.Comment: final version. Math. Z. (to appear

Fourier PCA and Robust Tensor Decomposition

Fourier PCA is Principal Component Analysis of a matrix obtained from higher order derivatives of the logarithm of the Fourier transform of a distribution.We make this method algorithmic by developing a tensor decomposition method for a pair of tensors sharing the same vectors in rank-$1$ decompositions. Our main application is the first provably polynomial-time algorithm for underdetermined ICA, i.e., learning an $n \times m$ matrix $A$ from observations $y=Ax$ where $x$ is drawn from an unknown product distribution with arbitrary non-Gaussian components. The number of component distributions $m$ can be arbitrarily higher than the dimension $n$ and the columns of $A$ only need to satisfy a natural and efficiently verifiable nondegeneracy condition. As a second application, we give an alternative algorithm for learning mixtures of spherical Gaussians with linearly independent means. These results also hold in the presence of Gaussian noise.Comment: Extensively revised; details added; minor errors corrected; exposition improve

Quantum and random walks as universal generators of probability distributions

Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without loosing coherence, as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.Comment: 8 pages, 2 figures; revised and enlarged versio

On Invariant Notions of Segre Varieties in Binary Projective Spaces

Invariant notions of a class of Segre varieties \Segrem(2) of PG(2^m - 1, 2) that are direct products of $m$ copies of PG(1, 2), $m$ being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains \Segrem(2) and is invariant under its projective stabiliser group \Stab{m}{2}. By embedding PG(2^m - 1, 2) into \PG(2^m - 1, 4), a basis of the latter space is constructed that is invariant under \Stab{m}{2} as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according as $m$ is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a \Stab{m}{2}-invariant geometric spread of lines of PG(2^m - 1, 2). This spread is also related with a \Stab{m}{2}-invariant non-singular Hermitian variety. The case $m=3$ is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under \Stab{3}{2}, while the points of PG(7, 2) form five orbits.Comment: 18 pages, 1 figure; v2 - version accepted in Designs, Codes and Cryptograph

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause