arXiv subjects

Qinxuan Pan

Publications and source records attributed to Qinxuan Pan.

5 recordsLinked to original sources

arXiv2020

Sample-Optimal and Efficient Learning of Tree Ising models

We show that $n$-variable tree-structured Ising models can be learned computationally-efficiently to within total variation distance $ε$ from an optimal $O(n \ln n/ε^2)$ samples, where $O(\cdot)$ hides an absolute constant which, importantly, does not depend on the model being learned - neither its tree nor the magnitude of its edge strengths, on which we place no assumptions. Our guarantees hold, in fact, for the celebrated Chow-Liu [1968] algorithm, using the plug-in estimator for estimating mutual information. While this (or any other) algorithm may fail to identify the structure of the underlying model correctly from a finite sample, we show that it will still learn a tree-structured model that is $ε$-close to the true one in total variation distance, a guarantee called "proper learning." Our guarantees do not follow from known results for the Chow-Liu algorithm and the ensuing literature on learning graphical models, including a recent renaissance of algorithms on this learning challenge, which only yield asymptotic consistency results, or sample-inefficient and/or time-inefficient algorithms, unless further assumptions are placed on the graphical model, such as bounds on the "strengths" of the model's edges/hyperedges. While we establish guarantees for a widely known and simple algorithm, the analysis that this algorithm succeeds and is sample-optimal is quite complex, requiring a hierarchical classification of the edges into layers with different reconstruction guarantees, depending on their strength, combined with delicate uses of the subadditivity of the squared Hellinger distance over graphical models to control the error accumulation.

cs.LG↗

arXiv2016

Square Hellinger Subadditivity for Bayesian Networks and its Applications to Identity Testing

We show that the square Hellinger distance between two Bayesian networks on the same directed graph, $G$, is subadditive with respect to the neighborhoods of $G$. Namely, if $P$ and $Q$ are the probability distributions defined by two Bayesian networks on the same DAG, our inequality states that the square Hellinger distance, $H^2(P,Q)$, between $P$ and $Q$ is upper bounded by the sum, $\sum_v H^2(P_{\{v\} \cup Π_v}, Q_{\{v\} \cup Π_v})$, of the square Hellinger distances between the marginals of $P$ and $Q$ on every node $v$ and its parents $Π_v$ in the DAG. Importantly, our bound does not involve the conditionals but the marginals of $P$ and $Q$. We derive a similar inequality for more general Markov Random Fields. As an application of our inequality, we show that distinguishing whether two Bayesian networks $P$ and $Q$ on the same (but potentially unknown) DAG satisfy $P=Q$ vs $d_{\rm TV}(P,Q)>ε$ can be performed from $\tilde{O}(|Σ|^{3/4(d+1)} \cdot n/ε^2)$ samples, where $d$ is the maximum in-degree of the DAG and $Σ$ the domain of each variable of the Bayesian networks. If $P$ and $Q$ are defined on potentially different and potentially unknown trees, the sample complexity becomes $\tilde{O}(|Σ|^{4.5} n/ε^2)$, whose dependence on $n, ε$ is optimal up to logarithmic factors. Lastly, if $P$ and $Q$ are product distributions over $\{0,1\}^n$ and $Q$ is known, the sample complexity becomes $O(\sqrt{n}/ε^2)$, which is optimal up to constant factors.

cs.LG↗

arXiv2014

A Counter-Example to Karlin's Strong Conjecture for Fictitious Play

Fictitious play is a natural dynamic for equilibrium play in zero-sum games, proposed by [Brown 1949], and shown to converge by [Robinson 1951]. Samuel Karlin conjectured in 1959 that fictitious play converges at rate $O(1/\sqrt{t})$ with the number of steps $t$. We disprove this conjecture showing that, when the payoff matrix of the row player is the $n \times n$ identity matrix, fictitious play may converge with rate as slow as $Ω(t^{-1/n})$.

cs.GT↗

arXiv2014

MITEoR: A Scalable Interferometer for Precision 21 cm Cosmology

We report on the MIT Epoch of Reionization (MITEoR) experiment, a pathfinder low-frequency radio interferometer whose goal is to test technologies that improve the calibration precision and reduce the cost of the high-sensitivity 3D mapping required for 21 cm cosmology. MITEoR accomplishes this by using massive baseline redundancy, which enables both automated precision calibration and correlator cost reduction. We demonstrate and quantify the power and robustness of redundancy for scalability and precision. We find that the calibration parameters precisely describe the effect of the instrument upon our measurements, allowing us to form a model that is consistent with $χ^2$ per degree of freedom < 1.2 for as much as 80% of the observations. We use these results to develop an optimal estimator of calibration parameters using Wiener filtering, and explore the question of how often and how finely in frequency visibilities must be reliably measured to solve for calibration coefficients. The success of MITEoR with its 64 dual-polarization elements bodes well for the more ambitious Hydrogen Epoch of Reionization Array (HERA) project and other next-generation instruments, which would incorporate many identical or similar technologies.

astro-ph.IM↗

arXiv2013

Mapping our Universe in 3D with MITEoR

Mapping our universe in 3D by imaging the redshifted 21 cm line from neutral hydrogen has the potential to overtake the cosmic microwave background as our most powerful cosmological probe, because it can map a much larger volume of our Universe, shedding new light on the epoch of reionization, inflation, dark matter, dark energy, and neutrino masses. We report on MITEoR, a pathfinder low-frequency radio interferometer whose goal is to test technologies that greatly reduce the cost of such 3D mapping for a given sensitivity. MITEoR accomplishes this by using massive baseline redundancy both to enable automated precision calibration and to cut the correlator cost scaling from N^2 to NlogN, where N is the number of antennas. The success of MITEoR with its 64 dual-polarization elements bodes well for the more ambitious HERA project, which would incorporate many identical or similar technologies using an order of magnitude more antennas, each with dramatically larger collecting area.

astro-ph.IM↗