Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation

In this paper we derive new uniform convergence estimates for the V-cycle MGM applied to symmetric positive definite Toeplitz block tridiagonal matrices, by also discussing few connections with previous results. More concretely, the contributions of this paper are as follows: (1) It tackles the Toeplitz systems directly for the elliptic PDEs. (2) Simple (traditional) restriction operator and prolongation operator are employed in order to handle general Toeplitz systems at each level of the recursion. Such a technique is then applied to systems of algebraic equations generated by the difference scheme of the two-dimensional fractional Feynman-Kac equation, which describes the joint probability density function of non-Brownian motion. In particular, we consider the two coarsening strategies, i.e., doubling the mesh size (geometric MGM) and Galerkin approach (algebraic MGM), which lead to the distinct coarsening stiffness matrices in the general case: however, several numerical experiments show that the two algorithms produce almost the same error behaviour.Comment: 26 page

A deep learning pipeline for product recognition on store shelves

Recognition of grocery products in store shelves poses peculiar challenges. Firstly, the task mandates the recognition of an extremely high number of different items, in the order of several thousands for medium-small shops, with many of them featuring small inter and intra class variability. Then, available product databases usually include just one or a few studio-quality images per product (referred to herein as reference images), whilst at test time recognition is performed on pictures displaying a portion of a shelf containing several products and taken in the store by cheap cameras (referred to as query images). Moreover, as the items on sale in a store as well as their appearance change frequently over time, a practical recognition system should handle seamlessly new products/packages. Inspired by recent advances in object detection and image retrieval, we propose to leverage on state of the art object detectors based on deep learning to obtain an initial productagnostic item detection. Then, we pursue product recognition through a similarity search between global descriptors computed on reference and cropped query images. To maximize performance, we learn an ad-hoc global descriptor by a CNN trained on reference images based on an image embedding loss. Our system is computationally expensive at training time but can perform recognition rapidly and accurately at test time

Analysis of attractor distances in Random Boolean Networks

We study the properties of the distance between attractors in Random Boolean Networks, a prominent model of genetic regulatory networks. We define three distance measures, upon which attractor distance matrices are constructed and their main statistic parameters are computed. The experimental analysis shows that ordered networks have a very clustered set of attractors, while chaotic networks' attractors are scattered; critical networks show, instead, a pattern with characteristics of both ordered and chaotic networks.Comment: 9 pages, 6 figures. Presented at WIRN 2010 - Italian workshop on neural networks, May 2010. To appear in a volume published by IOS Pres

Video Registration in Egocentric Vision under Day and Night Illumination Changes

With the spread of wearable devices and head mounted cameras, a wide range of application requiring precise user localization is now possible. In this paper we propose to treat the problem of obtaining the user position with respect to a known environment as a video registration problem. Video registration, i.e. the task of aligning an input video sequence to a pre-built 3D model, relies on a matching process of local keypoints extracted on the query sequence to a 3D point cloud. The overall registration performance is strictly tied to the actual quality of this 2D-3D matching, and can degrade if environmental conditions such as steep changes in lighting like the ones between day and night occur. To effectively register an egocentric video sequence under these conditions, we propose to tackle the source of the problem: the matching process. To overcome the shortcomings of standard matching techniques, we introduce a novel embedding space that allows us to obtain robust matches by jointly taking into account local descriptors, their spatial arrangement and their temporal robustness. The proposal is evaluated using unconstrained egocentric video sequences both in terms of matching quality and resulting registration performance using different 3D models of historical landmarks. The results show that the proposed method can outperform state of the art registration algorithms, in particular when dealing with the challenges of night and day sequences

Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol

This note is devoted to preconditioning strategies for non-Hermitian multilevel block Toeplitz linear systems associated with a multivariate Lebesgue integrable matrix-valued symbol. In particular, we consider special preconditioned matrices, where the preconditioner has a band multilevel block Toeplitz structure, and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices. In this respect, our main result is as follows. Let $I_k:=(-\pi,\pi)^k$, let $\mathcal M_s$ be the linear space of complex $s\times s$ matrices, and let $f,g:I_k\to\mathcal M_s$ be functions whose components $f_{ij},\,g_{ij}:I_k\to\mathbb C,\ i,j=1,\ldots,s,$ belong to $L^\infty$. Consider the matrices $T_n^{-1}(g)T_n(f)$, where $n:=(n_1,\ldots,n_k)$ varies in $\mathbb N^k$ and $T_n(f),T_n(g)$ are the multilevel block Toeplitz matrices of size $n_1\cdots n_ks$ generated by $f,g$. Then $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}\sim_\lambda g^{-1}f$, i.e. the family of matrices $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}$ has a global (asymptotic) spectral distribution described by the function $g^{-1}f$, provided $g$ possesses certain properties (which ensure in particular the invertibility of $T_n^{-1}(g)$ for all $n$) and the following topological conditions are met: the essential range of $g^{-1}f$, defined as the union of the essential ranges of the eigenvalue functions $\lambda_j(g^{-1}f),\ j=1,\ldots,s$, does not disconnect the complex plane and has empty interior. This result generalizes the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work, concerning the non-preconditioned case $g=1$. The last part of this note is devoted to numerical experiments, which confirm the theoretical analysis and suggest the choice of optimal GMRES preconditioning techniques to be used for the considered linear systems.Comment: 18 pages, 26 figure