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Wei Qian

Publications and source records attributed to Wei Qian.

65 records · Page 4Linked to original sources
arXiv2019

Global Convergence of Least Squares EM for Demixing Two Log-Concave Densities

This work studies the location estimation problem for a mixture of two rotation invariant log-concave densities. We demonstrate that Least Squares EM, a variant of the EM algorithm, converges to the true location parameter from a randomly initialized point. We establish the explicit convergence rates and sample complexity bounds, revealing their dependence on the signal-to-noise ratio and the tail property of the log-concave distribution. Moreover, we show that this global convergence property is robust under model mis-specification. Our analysis generalizes previous techniques for proving the convergence results for Gaussian mixtures. In particular, we make use of an angle-decreasing property for establishing global convergence of Least Squares EM beyond Gaussian settings, as $\ell_2$ distance contraction no longer holds globally for general log-concave mixtures.

stat.ML↗
arXiv2019

Global Convergence of EM Algorithm for Mixtures of Two Component Linear Regression

The Expectation-Maximization algorithm is perhaps the most broadly used algorithm for inference of latent variable problems. A theoretical understanding of its performance, however, largely remains lacking. Recent results established that EM enjoys global convergence for Gaussian Mixture Models. For Mixed Linear Regression, however, only local convergence results have been established, and those only for the high SNR regime. We show here that EM converges for mixed linear regression with two components (it is known that it may fail to converge for three or more), and moreover that this convergence holds for random initialization. Our analysis reveals that EM exhibits very different behavior in Mixed Linear Regression from its behavior in Gaussian Mixture Models, and hence our proofs require the development of several new ideas.

stat.ML↗
arXiv2019

Uniqueness of the welding problem for SLE and Liouville quantum gravity

We give a simple set of geometric conditions on curves $η$, $\tildeη$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $φ\colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $η$ with $φ(η) = \tildeη$ then $φ$ is a conformal automorphism of ${\mathbf H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $η$ is a non-space-filling SLE$_κ$ curve in ${\mathbf H}$ from $0$ to $\infty$ and $φ$ is a homeomorphism which is conformal on ${\mathbf H} \setminus η$ and $φ(η)$, $η$ are equal in distribution then $φ$ is a conformal automorphism of ${\mathbf H}$. Applying this result for $κ=4$ establishes that the welding operation for critical ($γ=2$) Liouville quantum gravity (LQG) is well-defined. Applying it for $κ\in (4,8)$ gives a new proof that the welding of two independent $κ/4$-stable looptrees of quantum disks to produce an SLE$_κ$ on top of an independent $4/\sqrtκ$-LQG surface is well-defined.

math.PR↗
arXiv2018

Conditioning a Brownian loop-soup cluster on a portion of its boundary

We show that if one conditions a cluster in a Brownian loop-soup $L$ (of any intensity) in a two-dimensional domain by a portion $l$ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $l$ satisfies the conformal restriction property while the other loops in $L$ form an independent loop-soup. This result holds when one discovers $l$ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among other things that a phase transition occurs at c = 14/15 for the connectedness of the loops that touch $l$. Our results can be viewed as an extension of some of the results in our earlier paper in the following two directions: There, a loop-soup cluster was conditioned on its entire outer boundary while we discover here only part of this boundary. And, while it was explained there that the strong decomposition using a Poisson point process of excursions that we derived there should be specific to the case of the critical loop-soup, we show here that in the subcritical cases, a weaker property involving the conformal restriction property nevertheless holds.

math.PR↗
arXiv2018

An Interactive Greedy Approach to Group Sparsity in High Dimensions

Sparsity learning with known grouping structure has received considerable attention due to wide modern applications in high-dimensional data analysis. Although advantages of using group information have been well-studied by shrinkage-based approaches, benefits of group sparsity have not been well-documented for greedy-type methods, which much limits our understanding and use of this important class of methods. In this paper, generalizing from a popular forward-backward greedy approach, we propose a new interactive greedy algorithm for group sparsity learning and prove that the proposed greedy-type algorithm attains the desired benefits of group sparsity under high dimensional settings. An estimation error bound refining other existing methods and a guarantee for group support recovery are also established simultaneously. In addition, we incorporate a general M-estimation framework and introduce an interactive feature to allow extra algorithm flexibility without compromise in theoretical properties. The promising use of our proposal is demonstrated through numerical evaluations including a real industrial application in human activity recognition at home. Supplementary materials for this article are available online.

stat.ML↗
arXiv2018

Coupling the Gaussian free fields with free and with zero boundary conditions via common level lines

We point out a new simple way to couple the Gaussian Free Field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample at random all the signs of the height gaps on its boundary-touching zero-level lines (these signs are alternating for the zero-boundary GFF) in order to obtain a free boundary GFF. Constructions and couplings of the free boundary GFF and its level lines via soups of reflected Brownian loops and their clusters are also discussed. Such considerations show for instance that in a domain with an axis of symmetry, if one looks at the overlay of a single usual Conformal Loop Ensemble CLE(3) with its own symmetric image, one obtains the CLE(4)-type collection of level lines of a GFF with mixed zero/free boundary conditions in the half-domain.

math.PR↗
arXiv2017

Decomposition of Brownian loop-soup clusters

We study the structure of Brownian loop-soup clusters in two dimensions. Among other things, we obtain the following decomposition of the clusters with critical intensity: When one conditions a loop-soup cluster by its outer boundary $γ$ (which is known to be an SLE(4)-type loop), then the union of all excursions away from $γ$ by all the Brownian loops in the loop-soup that touch $γ$ is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by $γ$. A related result that we derive and use is that the couplings of the Gaussian Free Field (GFF) with CLE(4) via level-lines (by Miller-Sheffield), of the square of the GFF with loop-soups via occupation times (by Le Jan), and of the CLE(4) with loop-soups via loop-soup clusters (by Sheffield and Werner) can be made to coincide. An instrumental role in our proof of this fact is played by Lupu's description of CLE(4) as limits of discrete loop-soup clusters.

math.PR↗
arXiv2017

Conformal restriction: The trichordal case

The study of conformal restriction properties in two-dimensions has been initiated by Lawler, Schramm and Werner who focused on the natural and important chordal case: They characterized and constructed all random subsets of a given simply connected domain that join two marked boundary points and that satisfy the additional restriction property. The radial case (sets joining an inside point to a boundary point) has then been investigated by Wu. In the present paper, we study the third natural instance of such restriction properties, namely the "trichordal case", where one looks at random sets that join three marked boundary points. This case involves somewhat more technicalities than the other two, as the construction of this family of random sets relies on special variants of SLE$_{8/3}$ processes with a drift term in the driving function that involves hypergeometric functions. It turns out that such a random set can not be a simple curve simultaneously in the neighborhood of all three marked points, and that the exponent $α= 20/27$ shows up in the description of the law of the skinniest possible symmetric random set with this trichordal restriction property.

math.PR↗
arXiv2016

Insurance Premium Prediction via Gradient Tree-Boosted Tweedie Compound Poisson Models

The Tweedie GLM is a widely used method for predicting insurance premiums. However, the structure of the logarithmic mean is restricted to a linear form in the Tweedie GLM, which can be too rigid for many applications. As a better alternative, we propose a gradient tree-boosting algorithm and apply it to Tweedie compound Poisson models for pure premiums. We use a profile likelihood approach to estimate the index and dispersion parameters. Our method is capable of fitting a flexible nonlinear Tweedie model and capturing complex interactions among predictors. A simulation study confirms the excellent prediction performance of our method. As an application, we apply our method to an auto insurance claim data and show that the new method is superior to the existing methods in the sense that it generates more accurate premium predictions, thus helping solve the adverse selection issue. We have implemented our method in a user-friendly R package that also includes a nice visualization tool for interpreting the fitted model.

stat.ME↗
arXiv2015

On the Forecast Combination Puzzle

It is often reported in forecast combination literature that a simple average of candidate forecasts is more robust than sophisticated combining methods. This phenomenon is usually referred to as the "forecast combination puzzle". Motivated by this puzzle, we explore its possible explanations including estimation error, invalid weighting formulas and model screening. We show that existing understanding of the puzzle should be complemented by the distinction of different forecast combination scenarios known as combining for adaptation and combining for improvement. Applying combining methods without consideration of the underlying scenario can itself cause the puzzle. Based on our new understandings, both simulations and real data evaluations are conducted to illustrate the causes of the puzzle. We further propose a multi-level AFTER strategy that can integrate the strengths of different combining methods and adapt intelligently to the underlying scenario. In particular, by treating the simple average as a candidate forecast, the proposed strategy is shown to avoid the heavy cost of estimation error and, to a large extent, solve the forecast combination puzzle.

stat.ME↗