Ngoc Mai Tran

Combinatorial types of tropical eigenvectors

The map which takes a square matrix to its tropical eigenvalue-eigenvector pair is piecewise linear. We determine the cones of linearity of this map. They are simplicial but they do not form a fan. Motivated by statistical ranking, we also study the restriction of that cone decomposition to the subspace of skew-symmetric matrices.

Size-biased permutation of a finite sequence with independent and identically distributed terms

This paper focuses on the size-biased permutation of

Robust Exponential Memory in Hopfield Networks

n

Polytropes and Tropical Eigenspaces: Cones of Linearity

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.

HodgeRank Is the Limit of Perron Rank

The Hopfield recurrent neural network is a classical auto-associative model of memory, in which collections of symmetrically coupled McCulloch–Pitts binary neurons interact to perform emergent computation. Although previous researchers have explored the potential of this network to solve combinatorial optimization problems or store reoccurring activity patterns 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 low-density error-correcting codes that achieve Shannon’s noisy channel 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.

Pairwise ranking: Choice of method can produce arbitrarily different rank order

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

Product-Mix Auctions and Tropical Geometry

\mathbb{R}^{n \times n}

Principal Component Analysis in an Asymmetric Norm

, whose face lattice is anti-isomorphic to the lattice of complete set of connected relations. This fan refines the non-fan partition of

Robust exponential binary pattern storage in Little-Hopfield networks

\mathbb{R}^{n \times n}

Tropical Geometry and Mechanism Design.

corresponding to cones of linearity of the eigenvector map. Our results answer open questions in a previous work with Sturmfels (Bull. Lond. Math. Soc. 54:27–36, 2013) and lead to a new combinatorial classification of polytropes and tropical eigenspaces.