A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geometry as a special case) is identical to Zhang's (2004) extension to the \(\alpha\)-geometry, where the name of the pair of monotone embedding functions \(\rho\) and \(\tau\) were used instead of \(F\) and \(H\) used in Harsha and Subrahamanian Moosath (2014). Their weighting function \(G\) for the Riemannian metric appears cosmetically due to a rewrite of the score function in log-representation as opposed to \((\rho, \tau)\)-representation in Zhang (2004). It is further shown here that the resulting metric and \(\alpha\)-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the \(\alpha\)-geometric structure. As a special case, Naudts' (2004) \(\phi\)-logarithm embedding (using the so-called \(\log_\phi\) function) is recovered with the identification \(\rho=\phi, \, \tau=\log_\phi\), with \(\phi\)-exponential \(\exp_\phi\) given by the associated convex function linking the two representations.

We study analytical and numerical properties of the L 1 - L 2 minimization problem for sparse representation of a signal over a highly coherent dictionary. Though the L 1 - L 2 metric is non-convex, it is Lipschitz continuous. The difference of convex algorithm (DCA) is readily applicable for computing the sparse representation coefficients. The L 1 - L 2 minimization appears as an initialization step of DCA. We further integrate DCA with a non-standard simulated annealing methodology to approximate globally sparse solutions. Non-Gaussian random perturbations are more effective than standard Gaussian perturbations for improving sparsity of solutions. In numerical experiments, we conduct an extensive comparison among sparse penalties such as L 1 - L 2 for L 1 - L 2 based on data from three specific applications (over-sampled discreet cosine basis, differential absorption optical spectroscopy, and image denoising) where

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In this paper, we try to clarify some of the questions related to a key concept in multivariate polynomial solving algorithm over a finite field: the degree of regularity. By the degree of regularity, here we refer to a concept first presented by Dubois and Gama, namely the lowest degree at which certain nontrivial degree drop of a polynomial system occurs. Currently, it is somehow commonly accepted that we can use this degree to estimate the complexity of solving a polynomial system, even though we do not have systematic empirical data or a theory to support such a claim. In this paper, we would like to clarify the situation with the help of experiments. We first define a concept of solving degree for a polynomial system. The key question we then need to clarify is the connection of solving degree and the degree of regularity with focus on quadratic systems. To exclude the cases that do not represent the general situation, we need to define when a system is degenerate and when it is irreducible. With extensive computer experiments, we show that the two concepts, the degree of regularity and the solving degree, are related for irreducible systems in the sense that the difference between the two degrees is indeed small, less than 3. But due to the limitation of our experiments, we speculate that this may not be the case for high degree cases.