arXiv subjects

Ngoc M. Tran

Publications and source records attributed to Ngoc M. Tran.

6 recordsLinked to original sources

arXiv2016

Iterated Gilbert Mosaics and Poisson Tropical Plane Curves

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.

math.PR↗

arXiv2015

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

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.

math.PR↗

arXiv2015

Robust exponential memory in Hopfield networks

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.

nlin.AO↗

arXiv2015

The Tropical Commuting Variety

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.

math.AG↗

arXiv2014

A binary Hopfield network with $1/\log(n)$ information rate and applications to grid cell decoding

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 ${1}/{\log(n)}$.

q-bio.NC↗

arXiv2013

Polytropes and Tropical Eigenspaces: Cones of Linearity

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.

math.CO↗