Partial Identifiability of Restricted Latent Class Models

Latent class models have wide applications in social and biological sciences. In many applications, pre-specified restrictions are imposed on the parameter space of latent class models, through a design matrix, to reflect practitioners' assumptions about how the observed responses depend on subjects' latent traits. Though widely used in various fields, such restricted latent class models suffer from non-identifiability due to their discreteness nature and complex structure of restrictions. This work addresses the fundamental identifiability issue of restricted latent class models by developing a general framework for strict and partial identifiability of the model parameters. Under correct model specification, the developed identifiability conditions only depend on the design matrix and are easily checkable, which provide useful practical guidelines for designing statistically valid diagnostic tests. Furthermore, the new theoretical framework is applied to establish, for the first time, identifiability of several designs from cognitive diagnosis applications

Superconductivity in a two-dimensional superconductor with Rashba and Dresselhaus spin-orbit couplings

We present a general model with both Rashba and Dresselhaus spin-orbit couplings to describe a two-dimensional noncentrosymmetric superconductor. The combined effects of the two spin-orbit couplings on superconductivity are investigated in the framework of mean-field theory. We find that the Rashba and Dresselhaus spin-orbit couplings result in similar effects on superconductivity if they are present solely in the system. Mixing of spin-singlet and triplet pairings in electron band is induced under the assumption that each quasiparticle band is p-wave paired. If the two types of spin-orbit couplings appear jointly, both the singlet and triplet pairings are weakened and decreased down to their minimum values in the equal-Rashba-Dresselhaus case.Comment: 5 pages, 4 figure

Learning Attribute Patterns in High-Dimensional Structured Latent Attribute Models

Structured latent attribute models (SLAMs) are a special family of discrete latent variable models widely used in social and biological sciences. This paper considers the problem of learning significant attribute patterns from a SLAM with potentially high-dimensional configurations of the latent attributes. We address the theoretical identifiability issue, propose a penalized likelihood method for the selection of the attribute patterns, and further establish the selection consistency in such an overfitted SLAM with diverging number of latent patterns. The good performance of the proposed methodology is illustrated by simulation studies and two real datasets in educational assessment

The Sufficient and Necessary Condition for the Identifiability and Estimability of the DINA Model

Cognitive Diagnosis Models (CDMs) are useful statistical tools in cognitive diagnosis assessment. However, as many other latent variable models, the CDMs often suffer from the non-identifiability issue. This work gives the sufficient and necessary condition for the identifiability of the basic DINA model, which not only addresses the open problem in Xu and Zhang (2016, Psychomatrika, 81:625-649) on the minimal requirement for the identifiability, but also sheds light on the study of more general CDMs, which often cover the DINA as a submodel. Moreover, we show the identifiability condition ensures the consistent estimation of the model parameters. From a practical perspective, the identifiability condition only depends on the Q-matrix structure and is easy to verify, which would provide a guideline for designing statistically valid and estimable cognitive diagnosis tests

Nonseparable Gaussian Stochastic Process: A Unified View and Computational Strategy

Gaussian stochastic process (GaSP) has been widely used as a prior over functions due to its flexibility and tractability in modeling. However, the computational cost in evaluating the likelihood is $O(n^3)$, where $n$ is the number of observed points in the process, as it requires to invert the covariance matrix. This bottleneck prevents GaSP being widely used in large-scale data. We propose a general class of nonseparable GaSP models for multiple functional observations with a fast and exact algorithm, in which the computation is linear ($O(n)$) and exact, requiring no approximation to compute the likelihood. We show that the commonly used linear regression and separable models are special cases of the proposed nonseparable GaSP model. Through the study of an epigenetic application, the proposed nonseparable GaSP model can accurately predict the genome-wide DNA methylation levels and compares favorably to alternative methods, such as linear regression, random forests and localized Kriging method

Constant Query Time (1+ϵ)(1 + \epsilon)-Approximate Distance Oracle for Planar Graphs

We give a $(1+\epsilon)$-approximate distance oracle with $O(1)$ query time for an undirected planar graph $G$ with $n$ vertices and non-negative edge lengths. For $\epsilon>0$ and any two vertices $u$ and $v$ in $G$, our oracle gives a distance $\tilde{d}(u,v)$ with stretch $(1+\epsilon)$ in $O(1)$ time. The oracle has size $O(n\log n ((\log n)/\epsilon+f(\epsilon)))$ and pre-processing time $O(n\log n((\log^3 n)/\epsilon^2+f(\epsilon)))$, where $f(\epsilon)=2^{O(1/\epsilon)}$. This is the first $(1+\epsilon)$-approximate distance oracle with $O(1)$ query time independent of $\epsilon$ and the size and pre-processing time nearly linear in $n$, and improves the query time $O(1/\epsilon)$ of previous $(1+\epsilon)$-approximate distance oracle with size nearly linear in $n$

Near-Linear Time Constant-Factor Approximation Algorithm for Branch-Decomposition of Planar Graphs

We give an algorithm which for an input planar graph $G$ of $n$ vertices and integer $k$, in $\min\{O(n\log^3n),O(nk^2)\}$ time either constructs a branch-decomposition of $G$ with width at most $(2+\delta)k$, $\delta>0$ is a constant, or a $(k+1)\times \lceil{\frac{k+1}{2}\rceil}$ cylinder minor of $G$ implying $bw(G)>k$, $bw(G)$ is the branchwidth of $G$. This is the first $\tilde{O}(n)$ time constant-factor approximation for branchwidth/treewidth and largest grid/cylinder minors of planar graphs and improves the previous $\min\{O(n^{1+\epsilon}),O(nk^2)\}$ ($\epsilon>0$ is a constant) time constant-factor approximations. For a planar graph $G$ and $k=bw(G)$, a branch-decomposition of width at most $(2+\delta)k$ and a $g\times \frac{g}{2}$ cylinder/grid minor with $g=\frac{k}{\beta}$, $\beta>2$ is constant, can be computed by our algorithm in $\min\{O(n\log^3n\log k),O(nk^2\log k)\}$ time.Comment: The mainly revision is the $O(nk^2)$ algorithm part (Section 4): added proofs for graphs with edge weights 1/2 and 1, and modified the proofs for finding the minimum separating cycle

On nonexistence and existence of positive global solutions to heat equation with a potential term on Riemannian manifolds

We reinvestigate nonexistence and existence of global positive solutions to heat equation with a potential term on Riemannian manifolds. Especially, we give a very natural sharp condition only in terms of the volume of geodesic ball to obtain nonexistence results.Comment: 25 page

Stochastic Nested Variance Reduction for Nonconvex Optimization

We study finite-sum nonconvex optimization problems, where the objective function is an average of $n$ nonconvex functions. We propose a new stochastic gradient descent algorithm based on nested variance reduction. Compared with conventional stochastic variance reduced gradient (SVRG) algorithm that uses two reference points to construct a semi-stochastic gradient with diminishing variance in each iteration, our algorithm uses $K+1$ nested reference points to build a semi-stochastic gradient to further reduce its variance in each iteration. For smooth nonconvex functions, the proposed algorithm converges to an $\epsilon$-approximate first-order stationary point (i.e., $\|\nabla F(\mathbf{x})\|_2\leq \epsilon$) within $\tilde{O}(n\land \epsilon^{-2}+\epsilon^{-3}\land n^{1/2}\epsilon^{-2})$ number of stochastic gradient evaluations. This improves the best known gradient complexity of SVRG $O(n+n^{2/3}\epsilon^{-2})$ and that of SCSG $O(n\land \epsilon^{-2}+\epsilon^{-10/3}\land n^{2/3}\epsilon^{-2})$. For gradient dominated functions, our algorithm also achieves a better gradient complexity than the state-of-the-art algorithms.Comment: 28 pages, 2 figures, 1 tabl

Control of the False Discovery Rate Under Arbitrary Covariance Dependence

Multiple hypothesis testing is a fundamental problem in high dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any genes are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In the current paper, we propose a new methodology based on principal factor approximation, which successfully substracts the common dependence and weakens significantly the correlation structure, to deal with an arbitrary dependence structure. We derive the theoretical distribution for false discovery proportion (FDP) in large scale multiple testing when a common threshold is used and provide a consistent FDP. This result has important applications in controlling FDR and FDP. Our estimate of FDP compares favorably with Efron (2007)'s approach, as demonstrated by in the simulated examples. Our approach is further illustrated by some real data applications.Comment: 44 pages, 7 figure