Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networks

Convolutional neural networks (CNNs) have been shown to achieve optimal approximation and estimation error rates (in minimax sense) in several function classes. However, previous analyzed optimal CNNs are unrealistically wide and difficult to obtain via optimization due to sparse constraints in important function classes, including the H\"older class. We show a ResNet-type CNN can attain the minimax optimal error rates in these classes in more plausible situations -- it can be dense, and its width, channel size, and filter size are constant with respect to sample size. The key idea is that we can replicate the learning ability of Fully-connected neural networks (FNNs) by tailored CNNs, as long as the FNNs have \textit{block-sparse} structures. Our theory is general in a sense that we can automatically translate any approximation rate achieved by block-sparse FNNs into that by CNNs. As an application, we derive approximation and estimation error rates of the aformentioned type of CNNs for the Barron and H\"older classes with the same strategy.Comment: 8 pages + References 2 pages + Supplemental material 18 page

Double-digest RADseq loci using standard Illumina indexes improve deep and shallow phylogenetic resolution of Lophodermium, a widespread fungal endophyte of pine needles.

The phylogenetic and population genetic structure of symbiotic microorganisms may correlate with important ecological traits that can be difficult to directly measure, such as host preferences or dispersal rates. This study develops and tests a low-cost double-digest restriction site-associated DNA sequencing (ddRADseq) protocol to reveal among- and within-species genetic structure for Lophodermium, a genus of fungal endophytes whose evolutionary analyses have been limited by the scarcity of informative markers. The protocol avoids expensive barcoded adapters and incorporates universal indexes for multiplexing. We tested for reproducibility and functionality by comparing shared loci from sample replicates and assessed the effects of numbers of ambiguous sites and clustering thresholds on coverage depths, number of shared loci among samples, and phylogenetic reconstruction. Errors between technical replicates were minimal. Relaxing the quality-filtering criteria increased the mean coverage depth per locus and the number of loci recovered within a sample, but had little effect on the number of shared loci across samples. Increasing clustering threshold decreased the mean coverage depth per cluster and increased the number of loci recovered within a sample but also decreased the number of shared loci across samples, especially among distantly related species. The combination of low similarity clustering (70%) and relaxed quality-filtering (allowing up to 30 ambiguous sites per read) performed the best in phylogenetic analyses at both recent and deep genetic divergences. Hence, this method generated sufficient number of shared homologous loci to investigate the evolutionary relationships among divergent fungal lineages with small haploid genomes. The greater genetic resolution also revealed new structure within species that correlated with ecological traits, providing valuable insights into their cryptic life histories

Selection, Stability and Renormalization

We illustrate how to extend the concept of structural stability through applying it to the front propagation speed selection problem. This consideration leads us to a renormalization group study of the problem. The study illustrates two very general conclusions: (1) singular perturbations in applied mathematics are best understood as renormalized perturbation methods, and (2) amplitude equations are renormalization group equations.Comment: 38 pages, LaTeX, two PostScript figures available by anonymous ftp to gijoe.mrl.uiuc.edu (128.174.119.153) files /pub/front_kkfest_fig

Comment on ``Phase ordering in chaotic map lattices with conserved dynamics''

Angelini, Pellicoro, and Stramaglia [Phys. Rev. E {\bf 60}, R5021 (1999), cond-mat/9907149] (APS) claim that the phase ordering of two-dimensional systems of sequentially-updated chaotic maps with conserved ``order parameter'' does not belong, for large regions of parameter space, to the expected universality class. We show here that these results are due to a slow crossover and that a careful treatment of the data yields normal dynamical scaling. Moreover, we construct better models, i.e. synchronously-updated coupled map lattices, which are exempt from these crossover effects, and allow for the first precise estimates of persistence exponents in this case.Comment: 3 pages, to be published in Phys. Rev.

Lamellae Stability in Confined Systems with Gravity

The microphase separation of a diblock copolymer melt confined by hard walls and in the presence of a gravitational field is simulated by means of a cell dynamical system model. It is found that the presence of hard walls normal to the gravitational field are key ingredients to the formation of well ordered lamellae in BCP melts. To this effect the currents in the directions normal and parallel to the field are calculated along the interface of a lamellar domain, showing that the formation of lamellae parallel to the hard boundaries and normal to the field correspond to the stable configuration. Also, it is found thet the field increases the interface width.Comment: 4 pages, 2 figures, submitted to Physical Review