Boris Hanin (TAMU)
Abstract: I will present some recent joint work with Mihai Nica (Toronto) and ongoing work with Grigoris Paouris (Texas A&M) about products of N independent matrices of size n x n in the regime where both the size n and the number of terms N tends to infinity. I will discuss in particular the work in progress with Paouris about the bulk distribution of singular values when N > polylog(n) and about the normality of the largest Lyapunov exponents when N > n.
A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph
F into a large network G. When G is the complete graph with q nodes, this becomes the well-known problem of sampling uniform q-colorings of F. We propose two complementary MCMC algorithms for sampling a random graph homomorphisms and establish bounds on their mixing times and concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neigborhood sampling. Various time averages of the MCMC trajectory give us real-, function-, and network-valued computable observables, including well-known ones such as homomorphism density and average clustering coefficient. One of the main observable we propose is called the conditional homomorphism density profile, which reveals hierarchical structure of the network. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We also provide various examples and simulations demonstrating our framework through synthetic and real-world networks. For instance, we apply our framework to analyze Word Adjacency Networks of a 45 novels data set and propose an authorship attribution scheme using motif sampling and conditional homomorphism density profiles.
We consider the problem of active linear regression with ell2-bounded noise. In this context, the learner receives a set of unlabeled data points, chooses a small subset to receive the labels for, and must give an estimate of the function that performs well on fresh samples. We give an algorithm that is simultaneously optimal in the number of labeled and unlabeled data points, with O(d) labeled samples; previous work required O(dlogd) labeled samples regardless of the number of unlabeled samples.
Our results also apply to learning linear functions from noisy queries, again achieving optimal sample complexities. The techniques extend beyond linear functions, giving improved sample complexities for learning the family of k-Fourier-sparse signals with continuous frequencies. This is joint work with Xue Chen.
Neural computations emerge from myriad neuronal interactions occurring in intricate spiking networks. Due to the inherent complexity of neural models, relating the spiking activity of a network to its structure requires simplifying assumptions, such as considering models in the thermodynamic mean-field limit. In the thermodynamic mean-field limit, an infinite number of neurons interact via vanishingly small interactions, thereby erasing the finite size of interactions. To better capture the finite-size effects of interactions, we propose to analyze the activity of neural networks in the replica-mean-field limit. Replica-mean-field models are made of infinitely many replicas which interact according to the same basic structure as that of the finite network of interest. Here, we analytically characterize the stationary dynamics of an intensity-based neural network with spiking reset and heterogeneous excitatory synapses in the replica-mean-field limit. Specifically, we functionally characterize the stationary dynamics of these limit networks via ordinary differential equations derived from the Poisson hypothesis of queuing theory. We then reduce this functional characterization to a system of self-consistency equations specifying the stationary neuronal firing rates. Of general applicability, our approach combines rate-conservation principles from point-process theory and analytical considerations from generating-function methods. We validate our approach by demonstrating numerically that replica-mean-field models better capture the dynamics of feedforward neural networks with large, sparse connections than their thermodynamic counterparts. Finally, we explain that improved performance by analyzing the neuronal rate-transfer functions, which saturate due to finite-size effects in the replica-mean-field limit. |

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