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Extreme value statistics is the max analogue of classical statistics, while tropical geometry is the max analogue of classical geometry. In this paper, we review recent work where insights from tropical geometry were used to develop new, efficient learning algorithms with leading performance on benchmark datasets in extreme value statistics. We give intuition, backed by performances on benchmark datasets, for why and when causal inference for extremes should be employed over classical methods. Finally, we list some open problems at the intersection of causal inference, tropical geometry and deep learning., Comment: 19 pages, 5 figures

In the past few years, the study of receptive field codes has been of large interest to mathematicians. Here we give a complete characterization of receptive field codes realizable by connected receptive fields and we give the minimal embedding dimensions of these codes. In particular, we show that all connected codes are realizable in dimension at most 3. To our knowledge, this is the first family of receptive field codes for which the exact characterization and minimal embedding dimension is known., Comment: 9 pages, 4 figures

We propose an iterated version of the Gilbert model, which results in a sequence of random mosaics of the plane. We prove that under appropriate scaling, this sequence of mosaics converges to that obtained by a classical Poisson line process with explicit cylindrical measure. Our model arises from considerations on tropical plane curves, which are zeros of random tropical polynomials in two variables. In particular, the iterated Gilbert model convergence allows one to derive a scaling limit for Poisson tropical plane curves. Our work raises a number of open questions at the intersection of stochastic and tropical geometry., Comment: 36 pages, 11 figures, with appendix

We study tropical commuting matrices from two viewpoints: linear algebra and algebraic geometry. In classical linear algebra, there exist various criteria to test whether two square matrices commute. We ask for similar criteria in the realm of tropical linear algebra, giving conditions for two tropical matrices that are polytropes to commute. From the algebro-geometric perspective, we explicitly compute the tropicalization of the classical variety of commuting matrices in dimension 2 and 3., Comment: 14 pages, 4 figures

The Hopfield recurrent neural network is a classical auto-associative model of memory, in which collections of symmetrically-coupled McCulloch-Pitts neurons interact to perform emergent computation. Although previous researchers have explored the potential of this network to solve combinatorial optimization problems and store memories as attractors of its deterministic dynamics, a basic open problem is to design a family of Hopfield networks with a number of noise-tolerant memories that grows exponentially with neural population size. Here, we discover such networks by minimizing probability flow, a recently proposed objective for estimating parameters in discrete maximum entropy models. By descending the gradient of the convex probability flow, our networks adapt synaptic weights to achieve robust exponential storage, even when presented with vanishingly small numbers of training patterns. In addition to providing a new set of error-correcting codes that achieve Shannon's channel capacity bound, these networks also efficiently solve a variant of the hidden clique problem in computer science, opening new avenues for real-world applications of computational models originating from biology., Comment: 23 pages, 6 figures

A Hopfield network is an auto-associative, distributive model of neural memory storage and retrieval. A form of error-correcting code, the Hopfield network can learn a set of patterns as stable points of the network dynamic, and retrieve them from noisy inputs -- thus Hopfield networks are their own decoders. Unlike in coding theory, where the information rate of a good code (in the Shannon sense) is finite but the cost of decoding does not play a role in the rate, the information rate of Hopfield networks trained with state-of-the-art learning algorithms is of the order ${\log(n)}/{n}$, a quantity that tends to zero asymptotically with $n$, the number of neurons in the network. For specially constructed networks, the best information rate currently achieved is of order ${1}/{\sqrt{n}}$. In this work, we design simple binary Hopfield networks that have asymptotically vanishing error rates at an information rate of ${1}/{\log(n)}$. These networks can be added as the decoders of any neural code with noisy neurons. As an example, we apply our network to a binary neural decoder of the grid cell code to attain information rate ${1}/{\log(n)}$., Comment: extended abstract, 4 pages, 2 figures

This paper focuses on the size-biased permutation of $n$ independent and identically distributed (i.i.d.) positive random variables. This is a finite dimensional analogue of the size-biased permutation of ranked jumps of a subordinator studied in Perman-Pitman-Yor (PPY) [Probab. Theory Related Fields 92 (1992) 21-39], as well as a special form of induced order statistics [Bull. Inst. Internat. Statist. 45 (1973) 295-300; Ann. Statist. 2 (1974) 1034-1039]. This intersection grants us different tools for deriving distributional properties. Their comparisons lead to new results, as well as simpler proofs of existing ones. Our main contribution, Theorem 25 in Section 6, describes the asymptotic distribution of the last few terms in a finite i.i.d. size-biased permutation via a Poisson coupling with its few smallest order statistics., Comment: Published at http://dx.doi.org/10.3150/14-BEJ652 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

The map which takes a square matrix $A$ to its polytrope is piecewise linear. We show that cones of linearity of this map form a polytopal fan partition of $\{R}^{n \times n}$, whose face lattice is anti-isomorphic to the lattice of complete set of connected relations. This fan refines the non-fan partition of $\R^{n \times n}$ corresponding to cones of linearity of the eigenvector map. Our results answer open questions in a previous work with Sturmfels and lead to a new combinatorial classification of polytropes and tropical eigenspaces., Comment: 18 pages, 7 figures, version 2 with proof of polytopality