We explore various sparse regularization techniques for analyzing fMRI data, such as the ℓ1 norm (often called LASSO in the context of a squared loss function), elastic net, and the recently introduced k-support norm. Employing sparsity regularization allows us to handle the curse of dimensionality, a problem commonly found in fMRI analysis. In this work we consider sparse regularization in bot...

In the convergence analysis of the GMRES method for a given matrix A, one quantity of interest is the largest possible residual norm that can be attained, at a given iteration step k, over all unit norm initial vectors. This quantity is called the worst-case GMRES residual norm for A and k. We show that the worst case behavior of GMRES for the matrices A and A is the same, and we analyze proper...

Sparsity regularization allows handling the curse of dimensionality, a problem commonly found in fMRI data. In this paper, we compare LASSO (`1 regularization) and the recently introduced k-support norm on their ability to predict real valued variables from brain fMRI data for cocaine addiction, in a principled model selection setting. Furthermore, in the context of these two regularization met...

The sparse representation based classifier (SRC) is a classical representation method for classification. The solution of SRC is obtained by l1 norm minimization, which can not obtain the closed form solution. Thus, the computational complexity of SRC is a little high. The collaborative representation classifier (CRC) is another classical method for classification. The solution of CRC is obtain...

Let D be a masa in B(H) where H is a separable Hilbert space. We find real numbers η0 < η1 < η2 < · · · < η6 so that for every bounded, normal D-bimodule map Φ on B(H), either ‖Φ‖ > η6 or ‖Φ‖ = ηk for some k ∈ {0, 1, 2, 3, 4, 5, 6}. When D is totally atomic, these maps are the idempotent Schur multipliers and we characterise those with norm ηk for 0 ≤ k ≤ 6. We also show that the Schur idempote...

We introduce a new way of approximating initial condition to the semidiscrete finite element method for integro-differential equations using any degree of elements. We obtain several superconvergence results for the error between the approximate solution and the Ritz-Volterra projection of the exact solution. For k > 1, we obtain first order gain in Lp(2≤ p ≤∞) norm, second order in W1,p(2≤ p ≤...

This paper develops a multilevel least-squares approach for the numerical solution of the complex scalar exterior Helmholtz equation. This second-order equation is first recast into an equivalent first-order system by introducing several “field” variables. A combination of scaled L and H norms is then applied to the residual of this system to create a least-squares functional. It is shown that,...

We demonstrate that a large class of first-order quantum phase transitions, namely, transitions in which the ground state energy per particle is continuous but its first order derivative has jump discontinuity, can be described as condensation space states. Given system having Hamiltonian $H=K+gV$, where $K$ and $V$ are two non commuting operators acting on states $\mathbb{F}$, we may always wr...

We show that, if A is a finite-dimensional *-simple associative algebra with involution (over the field K of real or complex numbers) whose hermitian part H( A, * > is of degree > 3 over its center, if B is a unital algebra with involution over 06, and if (I.11 is an algebra norm on H( A @ B, * 1, then there exists an algebra norm on A @ B whose restriction to H(A @ B, *> is equivalent to 11 . ...

It is shown that the Vitali convergence theorem remains valid for the -upper integral. Using this result we prove completeness of the space L( ) with respect to the k kp-upper norm for 1 p < 1 , describe convergence of its elements in terms of the space L( ) for 1 p < 1 , give a necessary and sufficient condition for a sequence from L( ) to converge in the k kp-upper norm to a function from L( ...