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# Upcoming Seminars

Let Z(t) be a Gaussian stationary function on the real line, and fix a level L>0.

We are interested in the asymptotic behavior of the persistence probability: P(T) = P( Z(t) > L, for all t in [0,T] ).

One would guess that for "nice processes", the behavior of P(T) should be exponential. For non-negative correlations this may be established via sub-additivity arguments. However, so far, not a single example with sign-changing correlations was known to exhibit existence of the limit of {Log P(T)}/T, as T approaches infinity (that is, to have a true "persistence exponent").

In the talk I will present a proof of existence of the persistence exponent, for processes whose spectral measure is monotone on [0,∞) and is continuous and non-vanishing at 0. This includes, for example, the sinc-kernel process (whose covariance function is sin(t)/t ).

Joint work with Ohad Feldheim and Sumit Mukherjee.

The past decade has witnessed the emergence of single-cell technologies that measure the expression level of genes at a single-cell resolution. These developments have revolutionized our understanding of the rich heterogeneity, structure, and dynamics of cellular populations, by probing the states of millions of cells, and their change under different conditions or over time. However, in standard experiments, information about the spatial context of cells, along with additional layers of information they encode about their location along dynamic processes (e.g. cell cycle or differentiation trajectories), is either lost or not explicitly accessible. This poses a fundamental problem for elucidating collective tissue function and mechanisms of cell-to-cell communication.

In this talk I will present computational approaches for addressing these challenges, by learning interpretable representations of structure, context and design principles for multicellular systems from single-cell information. I will first describe how the locations of cells in their tissue of origin and the resulting spatial gene expression can be probabilistically inferred from single-cell information by a generalized optimal-transport optimization framework, that can flexibly incorporate prior biological assumptions or knowledge derived from experiments. Inference in this case is based on an organization principle for spatial gene expression, namely a structural correspondence between distances of cells in expression and physical space, which we hypothesized and supported for different tissues. We used this framework to spatially reconstruct diverse tissues and organisms, including the fly embryo, mammalian intestinal epithelium and cerebellum, and further inferred spatially informative genes. Since cells encode multiple layers of information, in addition to their spatial context, I will also discuss several approaches for the disentanglement of single-cell gene expression into distinct biological processes, based on ideas rooted in random matrix theory and manifold learning. I will finally discuss how these results can be generalized to reveal principles underlying self-organization of cells into multicellular structures, setting the foundation for the computationally-directed design of cell-to-cell interactions optimized for specific tissue structure or function.

In this talk, we will discuss a new type of a pseudo-random object called a "pseudo-random pseudo-distribution". This object was introduced in the context of the BPL vs. L problem, and I will sketch a space-efficient construction of the latter for read-once branching programs that has near-optimal dependence on the error parameter. The talk is a distillation of a joint work with Mark Braverman and Sumegha Garg (the paper is available online: https://eccc.weizmann.ac.il/report/2017/161/).

In a joint work with Erez Lapid we constructed a new class of representations based on applying the RSK algorithm on Zelevinski's multisegments. Those constructions have the potential to be an alternative to the commonly used basis of standard representations. Intriguingly, this class also turned out to categorify a 45-year-old development in invariant theory: The Rota basis of standard bitableaux.

I will talk about this classical theme and its relation to representations of p-adic GL_n, as well the expected properties of our new class.

In the talk I will present the Hue-Net - a novel Deep Learning framework for Intensity-based Image-to-Image Translation.

The key idea is a new technique we term network augmentation which allows a __differentiable__ construction of intensity histograms from images.

We further introduce __differentiable__ representations of (__1D__) cyclic and joint (__2D__) histograms and use them for defining loss functions based on cyclic Earth Mover's Distance (__EMD__) and Mutual Information (MI). While the Hue-Net can be applied to several image-to-image translation tasks, we choose to demonstrate its strength on color transfer problems, where the aim is to paint a source image with the colors of a different target image. Note that the desired output image does not exist and therefore cannot be used for supervised pixel-to-pixel learning.

This is accomplished by using the __HSV__ color-space and defining an intensity-based loss that is built on the __EMD__ between the cyclic hue histograms of the output and the target images. To enforce color-free similarity between the source and the output images, we define a semantic-based loss by a __differentiable__ approximation of the MI of these images.

The incorporation of histogram loss functions in addition to an adversarial loss enables the construction of semantically meaningful and realistic images.

Promising results are presented for different __datasets__.

We develop differentially private mechanisms that achieve nearly instance-optimal losses, achieving lower loss than all appropriately unbiased mechanisms for any possible instance. We show that our mechanisms, with a modest increase in sample size (logarithmic or constant), are instance-optimal for a large family of functions. In contrast to existing mechanisms, which use the global or local sensitivity of the function being estimated, and so are necessarily instance suboptimal, the key to our construction is to use the inverse of the sensitivity. This allows a simple instance-optimal algorithm, and we develop several representative private mechanisms, including for the median and regression problems.

The rediscovery of the Archimedes Palimpsest in 1998 was a unique event: one of the most important authors of antiquity was read afresh, with many transformations made in the established text. Twenty years later, what have we learned from the Archimedes Palimpsest? What changed in our picture of the ancient history of mathematics?

Reviel Netz is Pat Suppes Professor of Greek Mathematics and Astronomy at Stanford University. Besides editing the Archimedes Palimpsest, he has published many books and articles on the history of Greek mathematics and on other subjects. His most recent book, "Scale, Space and Canon in Ancient Literary Culture", is now forthcoming from Cambridge University Press.

Refreshments will be served at 11:00am.

Numerous researchers recently applied empirical spectral analysis to the study of modern deep learning classifiers, observing spectral outliers and small but distinct bumps often seen beyond the edge of a "main bulk". This talk presents an important formal class/cross-class structure and shows how it lies at the origin of these visually striking patterns. The structure is shown to permeate the spectra of deepnet features, backpropagated errors, gradients, weights, Fisher Information matrix and Hessian, whether these are considered in the context of an individual layer or the concatenation of them all. The significance of the structure is illustrated by (i) demonstrating empirically that the feature class means separate gradually from the bulk as function of depth and become increasingly more orthogonal, (ii) proposing a correction to KFAC, a well known second-order optimization algorithm for training deepnets, and (ii) proving in the context of multinomial logistic regression that the ratio of outliers to bulk in the spectrum of the Fisher information matrix is predictive of misclassification.

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