arXiv subjects

Yuan Zhou

Publications and source records attributed to Yuan Zhou.

At least 289 records · Page 16Linked to original sources

arXiv2019

A quantative Sobolev regularity for absolute minimizers involving Hamiltonian $H(p)\in C^0 (\mathbb{R}^2)$ in plane

Suppose that $H \in C^0 (\mathbb{R}^2)$ satisfies \begin{enumerate} \item[(H1)] $H$ is locally strongly convex and locally strongly concave in $\rr^2$, \item[(H2)] $H(0)=\min_{p\in\rr^2}H(p)=0$. \end{enumerate} Let $Ω\subset \rr^2$ be any domain. For any $u$ absolute minimizer for $H$ in $Ω$, or if $H\in C^1(\rr^2)$ additionally, for any viscosity solution to the Aronsson equation $$\mathscr A_H[u]=\sum_{i,j=1}^2 H_{p_i}(Du) H_{p_j}(Du)u_{x_ix_j}=0 \quad \mbox{ in $Ω$,}$$ the following are proven in this paper: \begin{enumerate} \item[(i)] We have $[H(Du)]^α\in W^{1,2}_\loc(Ω)$ whenever $α>1/2-τ_H(0)$; some quantative upper bounds are also given. Here $τ_H(0)=1/2$ when $H\in C^2(\rr^2)$, and $0< τ_H(0)\le 1/2$ in general. \item[(ii)] If $H\in C^1(\rr^2)$, then the distributional determinant $-{\rm det}D^2u\,dx$ is a nonnegative Radon measure in $Ω$ and enjoys some quantative lower/upper bounds. \item[(iii)] If $H\in C^1(\rr^2)$, then for all $α>\frac12-τ_H(0)$, we have $$\mbox{$\langle D [H(Du )]^α,D_p H(Du )\rangle=0 $ almost everywhere in $Ω$}.$$ \end{enumerate}

math.AP↗

arXiv2019

Dynamic Assortment Optimization with Changing Contextual Information

In this paper, we study the dynamic assortment optimization problem under a finite selling season of length $T$. At each time period, the seller offers an arriving customer an assortment of substitutable products under a cardinality constraint, and the customer makes the purchase among offered products according to a discrete choice model. Most existing work associates each product with a real-valued fixed mean utility and assumes a multinomial logit choice (MNL) model. In many practical applications, feature/contexutal information of products is readily available. In this paper, we incorporate the feature information by assuming a linear relationship between the mean utility and the feature. In addition, we allow the feature information of products to change over time so that the underlying choice model can also be non-stationary. To solve the dynamic assortment optimization under this changing contextual MNL model, we need to simultaneously learn the underlying unknown coefficient and makes the decision on the assortment. To this end, we develop an upper confidence bound (UCB) based policy and establish the regret bound on the order of $\widetilde O(d\sqrt{T})$, where $d$ is the dimension of the feature and $\widetilde O$ suppresses logarithmic dependence. We further established the lower bound $Ω(d\sqrt{T}/K)$ where $K$ is the cardinality constraint of an offered assortment, which is usually small. When $K$ is a constant, our policy is optimal up to logarithmic factors. In the exploitation phase of the UCB algorithm, we need to solve a combinatorial optimization for assortment optimization based on the learned information. We further develop an approximation algorithm and an efficient greedy heuristic. The effectiveness of the proposed policy is further demonstrated by our numerical studies.

econ.EM↗

arXiv2019

Orlicz-Besov extension and Ahlfors $n$-regular domains

Let $n\ge2$ and $ϕ: [0,\fz) \to [0,\infty)$ be a Young's function satisfying $\sup_{x>0} \int_0^1\frac{ϕ( t x)}{ ϕ(x)}\frac{dt}{t^{n+1} }<\infty. $ We show that Ahlfors $n$-regular domains are Besov-Orlicz ${\dot {\bf B}}^ϕ$ extension domains, which is necessary to guarantee the nontrivially of ${\dot {\bf B}}^ϕ$. On the other hand, assume that $ϕ$ grows sub-exponentially at $\fz$ additionally. If $Ω$ is a Besov-Orlicz ${\dot {\bf B}}^ϕ$ extension domain, then it must be Ahlfors $n$-regular.

math.FA↗

arXiv2019

Regularity of absolute minimizers for continuous convex Hamiltonians

For any $n\ge 2$, $Ω\subset\rn$, and any given convex and coercive Hamiltonian function $H\in C^{0}(\rn)$, we find an optimal sufficient condition on $H$, that is, for any $c\in\mathbb R$, the level set $H^{-1}(c)$ does not contains any line segment, such then any absolute minimizer $u\in AM_H(Ω)$ enjoys the linear approximation property. As consequences, we show that when $n=2$, if $u\in AM_H(Ω)$ then $u\in C^1$; and if $u\in AM_H(\rr^2)$ satisfies a linear growth at the infinity, then $u$ is a linear function on $\rr^2$. In particular, if $H$ is a strictly convex Banach norm $\|\cdot\|$ on $\mathbb R^2$, e.g. the $l_α$-norm for $1<α<1$, then any $u\in AM_H(Ω)$ is $C^1$. The ideas of proof are, instead of PDE approaches, purely variational and geometric.

math.AP↗

arXiv2019

Hamiltonian Monte Carlo for Probabilistic Programs with Discontinuities

Hamiltonian Monte Carlo (HMC) is arguably the dominant statistical inference algorithm used in most popular "first-order differentiable" Probabilistic Programming Languages (PPLs). However, the fact that HMC uses derivative information causes complications when the target distribution is non-differentiable with respect to one or more of the latent variables. In this paper, we show how to use extensions to HMC to perform inference in probabilistic programs that contain discontinuities. To do this, we design a Simple first-order Probabilistic Programming Language (SPPL) that contains a sufficient set of language restrictions together with a compilation scheme. This enables us to preserve both the statistical and syntactic interpretation of if-else statements in the probabilistic program, within the scope of first-order PPLs. We also provide a corresponding mathematical formalism that ensures any joint density denoted in such a language has a suitably low measure of discontinuities.

stat.CO↗

arXiv2018

On Exploration, Exploitation and Learning in Adaptive Importance Sampling

We study adaptive importance sampling (AIS) as an online learning problem and argue for the importance of the trade-off between exploration and exploitation in this adaptation. Borrowing ideas from the bandits literature, we propose Daisee, a partition-based AIS algorithm. We further introduce a notion of regret for AIS and show that Daisee has $\mathcal{O}(\sqrt{T}(\log T)^{\frac{3}{4}})$ cumulative pseudo-regret, where $T$ is the number of iterations. We then extend Daisee to adaptively learn a hierarchical partitioning of the sample space for more efficient sampling and confirm the performance of both algorithms empirically.

stat.ML↗

arXiv2018

Tight Bounds for Collaborative PAC Learning via Multiplicative Weights

We study the collaborative PAC learning problem recently proposed in Blum et al.~\cite{BHPQ17}, in which we have $k$ players and they want to learn a target function collaboratively, such that the learned function approximates the target function well on all players' distributions simultaneously. The quality of the collaborative learning algorithm is measured by the ratio between the sample complexity of the algorithm and that of the learning algorithm for a single distribution (called the overhead). We obtain a collaborative learning algorithm with overhead $O(\ln k)$, improving the one with overhead $O(\ln^2 k)$ in \cite{BHPQ17}. We also show that an $Ω(\ln k)$ overhead is inevitable when $k$ is polynomial bounded by the VC dimension of the hypothesis class. Finally, our experimental study has demonstrated the superiority of our algorithm compared with the one in Blum et al. on real-world datasets.

cs.LG↗

arXiv2018

Preparing multiparticle entangled states of NV centers via adiabatic ground-state transitions

We propose an efficient method to generate multiparticle entangled states of NV centers in a spin mechanical system, where the spins interact through a collective coupling of the Lipkin-Meshkov-Glick (LMG) type. We show that, through adiabatic transitions in the ground state of the LMG Hamiltonian, the Greenberger-Horne-Zeilinger (GHZ)-type or the W-type entangled states of the NV spins can be generated with this hybrid system from an initial product state. Because of adiabaticity, this scheme is robust against practical noise and experimental imperfection, and may be useful for quantum information processing.

quant-ph↗

arXiv2018

Off-Policy Evaluation and Learning from Logged Bandit Feedback: Error Reduction via Surrogate Policy

When learning from a batch of logged bandit feedback, the discrepancy between the policy to be learned and the off-policy training data imposes statistical and computational challenges. Unlike classical supervised learning and online learning settings, in batch contextual bandit learning, one only has access to a collection of logged feedback from the actions taken by a historical policy, and expect to learn a policy that takes good actions in possibly unseen contexts. Such a batch learning setting is ubiquitous in online and interactive systems, such as ad platforms and recommendation systems. Existing approaches based on inverse propensity weights, such as Inverse Propensity Scoring (IPS) and Policy Optimizer for Exponential Models (POEM), enjoy unbiasedness but often suffer from large mean squared error. In this work, we introduce a new approach named Maximum Likelihood Inverse Propensity Scoring (MLIPS) for batch learning from logged bandit feedback. Instead of using the given historical policy as the proposal in inverse propensity weights, we estimate a maximum likelihood surrogate policy based on the logged action-context pairs, and then use this surrogate policy as the proposal. We prove that MLIPS is asymptotically unbiased, and moreover, has a smaller nonasymptotic mean squared error than IPS. Such an error reduction phenomenon is somewhat surprising as the estimated surrogate policy is less accurate than the given historical policy. Results on multi-label classification problems and a large- scale ad placement dataset demonstrate the empirical effectiveness of MLIPS. Furthermore, the proposed surrogate policy technique is complementary to existing error reduction techniques, and when combined, is able to consistently boost the performance of several widely used approaches.

cs.LG↗

arXiv2018

Inference Trees: Adaptive Inference with Exploration

We introduce inference trees (ITs), a new class of inference methods that build on ideas from Monte Carlo tree search to perform adaptive sampling in a manner that balances exploration with exploitation, ensures consistency, and alleviates pathologies in existing adaptive methods. ITs adaptively sample from hierarchical partitions of the parameter space, while simultaneously learning these partitions in an online manner. This enables ITs to not only identify regions of high posterior mass, but also maintain uncertainty estimates to track regions where significant posterior mass may have been missed. ITs can be based on any inference method that provides a consistent estimate of the marginal likelihood. They are particularly effective when combined with sequential Monte Carlo, where they capture long-range dependencies and yield improvements beyond proposal adaptation alone.

stat.CO↗

arXiv2018

Optimal Design of Process Flexibility for General Production Systems

Process flexibility is widely adopted as an effective strategy for responding to uncertain demand. Many algorithms for constructing sparse flexibility designs with good theoretical guarantees have been developed for balanced and symmetrical production systems. These systems assume that the number of plants equals the number of products, that supplies have the same capacity, and that demands are independently and identically distributed. In this paper, we relax these assumptions and consider a general class of production systems. We construct a simple flexibility design to fulfill $(1-ε)$-fraction of expected demand with high probability (w.h.p.) where the average degree is $O(\ln(1/ε))$. To motivate our construction, we first consider a natural weighted probabilistic construction from Chou et al. (2011) where the degree of each node is proportional to its expected capacity. However, this strategy is shown to be sub-optimal. To obtain an optimal construction, we develop a simple yet effective thresholding scheme. The analysis of our approach extends the classical analysis of expander graphs by overcoming several technical difficulties. Our approach may prove useful in other applications that require expansion properties of graphs with non-uniform degree sequences.

cs.DS↗

arXiv2018

An asymtotic sharp Sobolev regularity for planar infinity harmonic functions

Given an arbitrary planar $\infty$-harmonic function $u$, for each $α>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^α$, which is sharp as $α\to0$. We also show that the distributional determinant of $u$ is a Radon measure enjoying some quantitative lower and upper bounds. As a by-product, for each $p>2$ we obtain some quantitative local $W^{1,p}$-estimates of $u$, and consequently, an $L^p$-Liouville property for $\infty$-harmonic functions in whole plane.

math.AP↗

arXiv2018

Some sharp Sobolev regularity for inhomogeneous $\infty$-Laplace equation in plane

Suppose $Ω\Subset \mathbb R^2$ and $f\in BV_{loc}(Ω)\cap C^0(Ω)$ with $|f|>0$ in $Ω$. Let $u\in C^0(Ω)$ be a viscosity solution to the inhomogeneous $\infty$-Laplace equation $$ -Δ_{\infty} u :=-\frac12\sum_{i=1}^2(|Du|^2)_iu_i= -\sum_{i,j=1}^2u_iu_ju_{ij} =f \quad {\rm in}\ Ω. $$ The following are proved in this paper. (i) For $ α> 3/2$, we have $|Du|^α\in W^{1,2}_{loc}(Ω)$, which is (asymptotic) sharp when $ α\to 3/2$. Indeed, the function $w(x_1,x_2)=-x_1^ {4/3} $ is a viscosity solution to $-Δ_\infty w=\frac{4^3}{3^4}$ in $\mathbb R^2$. For any $p> 2$, $|Dw|^α\notin W^{1,p}_{loc}(\mathbb R^2)$ whenever $α\in(3/2,3-3/p)$. (ii) For $ α\in(0, 3/2]$ and $p\in[1, 3/(3-α))$, we have $|Du|^α\in W^{1,p}_{loc}(Ω)$, which is sharp when $p\to 3/(3-α)$. Indeed, $ |Dw|^α\notin W^{1,3/(3-α)}_{loc}(\mathbb R^2)$. (iii) For $ ε> 0$, we have $|Du|^{-3+ε}\in L^1_{loc}(Ω)$, which is sharp when $ε\to0$. Indeed, $|Dw|^{-3} \notin L^1_{loc}(\mathbb R^2)$. (iv) For $ α> 0$, we have $$-(|Du|^α)_iu_i= 2α|Du|^{ α-2}f \ \mbox{ almost everywhere in $Ω$}.$$ Some quantative bounds are also given.

math.AP↗

arXiv2018

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VI. The curious case of two-sided discontinuous minimal valid functions

We construct a two-sided discontinuous piecewise linear minimal valid function for the 1-row Gomory--Johnson model which is not extreme, but which is not a convex combination of other piecewise linear minimal valid functions. The new function only admits piecewise microperiodic perturbations. We present an algorithm for computations with a restricted class of such perturbations.

math.OC↗

arXiv2018

Unmixing urban hyperspectral imagery with a Gaussian mixture model on endmember variability

In this paper, we model a pixel as a linear combination of endmembers sampled from probability distributions of Gaussian mixture models (GMM). The parameters of the GMM distributions are estimated using spectral libraries. Abundances are estimated based on the distribution parameters. The advantage of this algorithm is that the model size grows very slowly as a function of the library size. To validate this method, we used data collected by the AVIRIS sensor over the Santa Barbara region: two 16 m spatial resolution and two 4 m spatial resolution images. 64 validated regions of interest (ROI) (180 m by 180 m) were used to assess estimate accuracy. Ground truth was obtained using 1 m images leading to the following 6 classes: turfgrass, non-photosynthetic vegetation (NPV), paved, roof, soil, and tree. Spectral libraries were built by manually identifying and extracting pure spectra from both resolution images, resulting in 3,287 spectra at 16 m and 15,426 spectra at 4 m. We then unmixed ROIs of each resolution using the following unmixing algorithms: the set-based algorithms MESMA and AAM, and the distribution-based algorithms GMM, NCM, and BCM. The original libraries were used for the distribution-based algorithms whereas set-based methods required a sophisticated reduction method, resulting in reduced libraries of 61 spectra at 16 m and 95 spectra at 4 m. The results show that GMM performs best among the distribution-based methods, producing comparable accuracy to MESMA, and may be more robust across datasets.

cs.CV↗

arXiv2018

A Gaussian mixture model representation of endmember variability in hyperspectral unmixing

Hyperspectral unmixing while considering endmember variability is usually performed by the normal compositional model (NCM), where the endmembers for each pixel are assumed to be sampled from unimodal Gaussian distributions. However, in real applications, the distribution of a material is often not Gaussian. In this paper, we use Gaussian mixture models (GMM) to represent the endmember variability. We show, given the GMM starting premise, that the distribution of the mixed pixel (under the linear mixing model) is also a GMM (and this is shown from two perspectives). The first perspective originates from the random variable transformation and gives a conditional density function of the pixels given the abundances and GMM parameters. With proper smoothness and sparsity prior constraints on the abundances, the conditional density function leads to a standard maximum a posteriori (MAP) problem which can be solved using generalized expectation maximization. The second perspective originates from marginalizing over the endmembers in the GMM, which provides us with a foundation to solve for the endmembers at each pixel. Hence, our model can not only estimate the abundances and distribution parameters, but also the distinct endmember set for each pixel. We tested the proposed GMM on several synthetic and real datasets, and showed its potential by comparing it to current popular methods.

cs.CV↗

arXiv2017

Simulating the Lipkin-Meshkov-Glick model in a hybrid quantum system

We propose an efficient scheme for simulating the Lipkin-Meshkov-Glick (LMG) model with nitrogen-vacancy (NV) center ensembles in diamond magnetically coupled to superconducting coplanar waveguide cavities. With the assistance of external microwave driving fields, we show that the interaction of the NV spins can be easily controlled, and several types of the LMG model can be realized by tuning the different parameters. Under the thermal dynamical limit, the distinct non-equilibrium second order quantum phase transition of the spin ensemble can be achieved at the critical point. Furthermore, we show that the spin squeezed state can be generated by tailoring the LMG Hamiltonian to possess the two-axis counter-twisting form in this hybrid quantum system.

quant-ph↗

arXiv2017

Variational Monte Carlo study of spin dynamics in underdoped cuprates

The hour-glass-like dispersion of spin excitations is a common feature of underdoped cuprates. It was qualitatively explained by the random phase approximation based on various ordered states with some phenomenological parameters; however, its origin remains elusive. Here, we present a numerical study of spin dynamics in the $t$-$J$ model using the variational Monte Carlo method. This parameter-free method satisfies the no double-occupancy constraint of the model and thus provides a better evaluation on the spin dynamics with respect to various mean-field trial states. We conclude that the lower branch of the hour-glass dispersion is a collective mode and the upper branch is more likely the consequence of the stripe state than the other candidates.

cond-mat.str-el↗