arXiv subjects
Publications and source records attributed to Holden Lee.
We give a polynomial-time algorithm for learning latent-state linear dynamical systems without system identification, and without assumptions on the spectral radius of the system's transition matrix. The algorithm extends the recently introduced technique of spectral filtering, previously applied only to systems with a symmetric transition matrix, using a novel convex relaxation to allow for the efficient identification of phases.
A key task in Bayesian statistics is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). However, without any assumptions, sampling (even approximately) can be #P-hard, and few works have provided "beyond worst-case" guarantees for such settings. For log-concave distributions, classical results going back to Bakry and Émery (1985) show that natural continuous-time Markov chains called Langevin diffusions mix in polynomial time. The most salient feature of log-concavity violated in practice is uni-modality: commonly, the distributions we wish to sample from are multi-modal. In the presence of multiple deep and well-separated modes, Langevin diffusion suffers from torpid mixing. We address this problem by combining Langevin diffusion with simulated tempering. The result is a Markov chain that mixes more rapidly by transitioning between different temperatures of the distribution. We analyze this Markov chain for the canonical multi-modal distribution: a mixture of gaussians (of equal variance). The algorithm based on our Markov chain provably samples from distributions that are close to mixtures of gaussians, given access to the gradient of the log-pdf. For the analysis, we use a spectral decomposition theorem for graphs (Gharan and Trevisan, 2014) and a Markov chain decomposition technique (Madras and Randall, 2002).
An open problem in complexity theory is to find the minimal degree of a polynomial representing the $n$-bit OR function modulo composite $m$. This problem is related to understanding the power of circuits with $\text{MOD}_m$ gates where $m$ is composite. The OR function is of particular interest because it is the simplest function not amenable to bounds from communication complexity. Tardos and Barrington established a lower bound of $Ω((\log n)^{O_m(1)})$, and Barrington, Beigel, and Rudich established an upper bound of $n^{O_m(1)}$. No progress has been made on closing this gap for twenty years, and progress will likely require new techniques. We make progress on this question viewed from a different perspective: rather than fixing the modulus $m$ and bounding the minimum degree $d$ in terms of the number of variables $n$, we fix the degree $d$ and bound $n$ in terms of the modulus $m$. For degree $d=2$, we prove a quasipolynomial bound of $n\le m^{O(d)}\le m^{O(\log m)}$, improving the previous best bound of $2^{O(m)}$ implied by Tardos and Barrington's general bound. To understand the computational power of quadratic polynomials modulo $m$, we introduce a certain dichotomy which may be of independent interest. Namely, we define a notion of boolean rank of a quadratic polynomial $f$ and relate it to the notion of diagonal rigidity. Using additive combinatorics, we show that when the rank is low, $f(\mathbf x)=0$ must have many solutions. Using techniques from exponential sums, we show that when the rank of $f$ is high, $f$ is close to equidistributed. In either case, $f$ cannot represent the OR function in many variables.
Folsom, Kent, and Ono used the theory of modular forms modulo $\ell$ to establish remarkable ``self-similarity'' properties of the partition function and give an overarching explanation of many partition congruences. We generalize their work to analyze powers $p_r$ of the partition function as well as Andrews's spt-function. By showing that certain generating functions reside in a small space made up of reductions of modular forms, we set up a general framework for congruences for $p_r$ and spt on arithmetic progressions of the form $\ell^mn+δ$ modulo powers of $\ell$. Our work gives a conceptual explanation of the exceptional congruences of $p_r$ observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.