Phase Transitions in Approximate Ranking

We study the problem of approximate ranking from observations of pairwise interactions. The goal is to estimate the underlying ranks of $n$ objects from data through interactions of comparison or collaboration. Under a general framework of approximate ranking models, we characterize the exact optimal statistical error rates of estimating the underlying ranks. We discover important phase transition boundaries of the optimal error rates. Depending on the value of the signal-to-noise ratio (SNR) parameter, the optimal rate, as a function of SNR, is either trivial, polynomial, exponential or zero. The four corresponding regimes thus have completely different error behaviors. To the best of our knowledge, this phenomenon, especially the phase transition between the polynomial and the exponential rates, has not been discovered before

Breathing mode of two-dimensional atomic Fermi gases in harmonic traps

For two-dimensional (2D) atomic Fermi gases in harmonic traps, the SO(2,1) symmetry is broken by the interatomic interaction explicitly via the contact correlation operator. Consequently the frequency of the breathing mode $\omega_B$ of the 2D Fermi gas can be different from $2\omega_0$, with $\omega_0$ the trapping frequency of harmonic potentials. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for $\omega_B$. We further calculate $\omega_B$ through the Euler equations in the hydrodynamic regime. The obtained value of $\omega_B$ satisfies the upper bounds and shows deviation from $2\omega_0$ which can be as large as about 8%.Comment: 5 pages, 1 figur

Minimax Optimal Convergence Rates for Estimating Ground Truth from Crowdsourced Labels

Crowdsourcing has become a primary means for label collection in many real-world machine learning applications. A classical method for inferring the true labels from the noisy labels provided by crowdsourcing workers is Dawid-Skene estimator. In this paper, we prove convergence rates of a projected EM algorithm for the Dawid-Skene estimator. The revealed exponent in the rate of convergence is shown to be optimal via a lower bound argument. Our work resolves the long standing issue of whether Dawid-Skene estimator has sound theoretical guarantees besides its good performance observed in practice. In addition, a comparative study with majority voting illustrates both advantages and pitfalls of the Dawid-Skene estimator

Three Identical Fermions with Resonant p-wave Interactions in Two Dimensions

A new kind of "super-Efimov" states of binding energies scaling as $\ln|E_n|\sim-e^{3n\pi/4}$ were predicted by a field theory calculation for three fermions with resonant $p$-wave interactions in two dimensions [Phys. Rev. Lett. \textbf{110}, 235301 (2013)]. However, the universality of these "super-Efimov" states has not been proved independently. In this Letter, we study the three fermion system through the hyperspherical formalism. Within the adiabatic approximation, we find that at $p$-wave resonances, the low energy physics of states of angular momentum $\ell=\pm1$ crucially depends on the value of an emergent dimensionless parameter $Y$ determined by the detail of the inter-particle potential. Only if $Y$ is exactly zero, the predicted "super-Efimov" states exist. If $Y>0$, the scaling of the bound states changes to $\ln|E_n|\sim-(n\pi)^2/2Y$, while there are no shallow bound states if $Y<0$.Comment: The neglected term $Q_{00}$ has been calculated numerically and found to be non-negligible in the large hyperradius limi

Steady-state phase diagram of quantum gases in a lattice coupled to a membrane

In a recent experiment [Vochezer {\it et al.,} Phys. Rev. Lett. \textbf{120}, 073602 (2018)], a novel kind of hybrid atom-opto-mechanical system has been realized by coupling atoms in a lattice to a membrane. While such system promises a viable contender in the competitive field of simulating non-equilibrium many-body physics, its complete steady-state phase diagram is still lacking. Here we study the phase diagram of this hybrid system based on an atomic Bose-Hubbard model coupled to a quantum harmonic oscillator. We take both the expectation value of the bosonic operator and the mechanical motion of the membrane as order parameters, and thereby identify four different quantum phases. Importantly, we find the atomic gas in the steady state of such non-equilibrium setting undergoes a superfluid-Mott-insulator transition when the atom-membrane coupling is tuned to increase. Such steady-state phase transition can be seen as the non-equilibrium counterpart of the conventional superfluid-Mott-insulator transition in the ground state of Bose-Hubbard model. Further, no matter which phase the quantum gas is in, the mechanic motion of the membrane exhibits a transition from an incoherent vibration to a coherent one when the atom-membrane coupling increases, agreeing with the experimental observations. Our present study provides a simple way to study non-equilibrium many-body physics that is complementary to ongoing experiments on the hybrid atomic and opto-mechanical systems.Comment: 7 pages, 3 figures; v2: Fig1(a) replace

Density Estimation with Contaminated Data: Minimax Rates and Theory of Adaptation

This paper studies density estimation under pointwise loss in the setting of contamination model. The goal is to estimate $f(x_0)$ at some $x_0\in\mathbb{R}$ with i.i.d. observations, $$X_1,\dots,X_n\sim (1-\epsilon)f+\epsilon g,$$ where $g$ stands for a contamination distribution. In the context of multiple testing, this can be interpreted as estimating the null density at a point. We carefully study the effect of contamination on estimation through the following model indices: contamination proportion $\epsilon$, smoothness of target density $\beta_0$, smoothness of contamination density $\beta_1$, and level of contamination $m$ at the point to be estimated, i.e. $g(x_0)\leq m$. It is shown that the minimax rate with respect to the squared error loss is of order $$[n^{-\frac{2\beta_0}{2\beta_0+1}}]\vee[\epsilon^2(1\wedge m)^2]\vee[n^{-\frac{2\beta_1}{2\beta_1+1}}\epsilon^{\frac{2}{2\beta_1+1}}],$$ which characterizes the exact influence of contamination on the difficulty of the problem. We then establish the minimal cost of adaptation to contamination proportion, to smoothness and to both of the numbers. It is shown that some small price needs to be paid for adaptation in any of the three cases. Variations of Lepski's method are considered to achieve optimal adaptation. The problem is also studied when there is no smoothness assumption on the contamination distribution. This setting that allows for an arbitrary contamination distribution is recognized as Huber's $\epsilon$-contamination model. The minimax rate is shown to be $$[n^{-\frac{2\beta_0}{2\beta_0+1}}]\vee [\epsilon^{\frac{2\beta_0}{\beta_0+1}}].$$ The adaptation theory is also different from the smooth contamination case. While adaptation to either contamination proportion or smoothness only costs a logarithmic factor, adaptation to both numbers is proved to be impossible

Convergence Rates of Variational Posterior Distributions

We study convergence rates of variational posterior distributions for nonparametric and high-dimensional inference. We formulate general conditions on prior, likelihood, and variational class that characterize the convergence rates. Under similar "prior mass and testing" conditions considered in the literature, the rate is found to be the sum of two terms. The first term stands for the convergence rate of the true posterior distribution, and the second term is contributed by the variational approximation error. For a class of priors that admit the structure of a mixture of product measures, we propose a novel prior mass condition, under which the variational approximation error of the mean-field class is dominated by convergence rate of the true posterior. We demonstrate the applicability of our general results for various models, prior distributions and variational classes by deriving convergence rates of the corresponding variational posteriors

Testing Network Structure Using Relations Between Small Subgraph Probabilities

We study the problem of testing for structure in networks using relations between the observed frequencies of small subgraphs. We consider the statistics \begin{align*} T_3 & =(\text{edge frequency})^3 - \text{triangle frequency}\\ T_2 & =3(\text{edge frequency})^2(1-\text{edge frequency}) - \text{V-shape frequency} \end{align*} and prove a central limit theorem for $(T_2, T_3)$ under an Erd\H{o}s-R\'{e}nyi null model. We then analyze the power of the associated $\chi^2$ test statistic under a general class of alternative models. In particular, when the alternative is a $k$-community stochastic block model, with $k$ unknown, the power of the test approaches one. Moreover, the signal-to-noise ratio required is strictly weaker than that required for community detection. We also study the relation with other statistics over three-node subgraphs, and analyze the error under two natural algorithms for sampling small subgraphs. Together, our results show how global structural characteristics of networks can be inferred from local subgraph frequencies, without requiring the global community structure to be explicitly estimated

Model Repair: Robust Recovery of Over-Parameterized Statistical Models

A new type of robust estimation problem is introduced where the goal is to recover a statistical model that has been corrupted after it has been estimated from data. Methods are proposed for "repairing" the model using only the design and not the response values used to fit the model in a supervised learning setting. Theory is developed which reveals that two important ingredients are necessary for model repair---the statistical model must be over-parameterized, and the estimator must incorporate redundancy. In particular, estimators based on stochastic gradient descent are seen to be well suited to model repair, but sparse estimators are not in general repairable. After formulating the problem and establishing a key technical lemma related to robust estimation, a series of results are presented for repair of over-parameterized linear models, random feature models, and artificial neural networks. Simulation studies are presented that corroborate and illustrate the theoretical findings

Spin-Orbit Coupled Spinor Bose-Einstein Condensates

An effective spin-orbit coupling can be generated in cold atom system by engineering atom-light interactions. In this letter we study spin-1/2 and spin-1 Bose-Einstein condensates with Rashba spin-orbit coupling, and find that the condensate wave function will develop non-trivial structures. From numerical simulation we have identified two different phases. In one phase the ground state is a single plane wave, and often we find the system splits into domains and an array of vortices plays the role as domain wall. In this phase, time-reversal symmetry is broken. In the other phase the condensate wave function is a standing wave and it forms spin stripe. The transition between them is driven by interactions between bosons. We also provide an analytical understanding of these results and determines the transition point between the two phases.Comment: 5 pages, 4 figure