Dropout and its extensions (e.g. DropBlock and DropConnect) are popular heuristics for training neural networks, which have been shown to improve generalization performance in practice. However, a theoretical understanding of their optimization and regularization
properties remains elusive. This talk will present a theoretical analysis of several dropout-like regularization strategies, all of which can be understood as stochastic gradient descent methods for minimizing a certain regularized loss. In the case of single
hidden-layer linear networks, we will show that Dropout and DropBlock induce nuclear norm and spectral k-support norm regularization, respectively, which promote solutions that are low-rank and balanced (i.e. have factors with equal norm). We will also show
that the global minimizer for Dropout and DropBlock can be computed in closed form, and that DropConnect is equivalent to Dropout. We will then show that some of these results can be extended to a general class of Dropout-strategies, and, with some assumptions,
to deep non-linear networks when Dropout is applied to the last layer.

In addition to frequent single-nucleotide mutations, mammalian and many
other genomes undergo rare and dramatic changes called genome
rearrangements. These include inversions, fissions, fusions, and
translocations. Although analysis of genome rearrangements was pioneered
by Dobzhansky and Sturtevant in 1938, we still know very little about the
rearrangement events that produced the existing varieties of genomic
architectures. Recovery of mammalian rearrangement history is a difficult
combinatorial problem that I will cover in this talk. Our data sets have
included sequenced genomes (human, mouse, rat, and others), as well as
radiation hybrid maps of additional mammals.

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.
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As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.
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The schedule of the lecture series will be approximately as follows:
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1. Heat Kernel and entropy estimates on Ricci flow backgrounds and related geometric bounds.
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2. Continuation of Lecture 1 + Synthetic definition of Ricci flows (metric flows) and basic properties
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3. Convergence and compactness theory of metric flows
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4. Partial regularity of limits of Ricci flows