arXiv subjects

Jingbo Liu

Publications and source records attributed to Jingbo Liu.

At least 37 records · Page 2Linked to original sources

arXiv2024

Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions

We study the asymptotic error of score-based diffusion model sampling in large-sample scenarios from a non-parametric statistics perspective. We show that a kernel-based score estimator achieves an optimal mean square error of $\widetilde{O}\left(n^{-1} t^{-\frac{d+2}{2}}(t^{\frac{d}{2}} \vee 1)\right)$ for the score function of $p_0*\mathcal{N}(0,t\boldsymbol{I}_d)$, where $n$ and $d$ represent the sample size and the dimension, $t$ is bounded above and below by polynomials of $n$, and $p_0$ is an arbitrary sub-Gaussian distribution. As a consequence, this yields an $\widetilde{O}\left(n^{-1/2} t^{-\frac{d}{4}}\right)$ upper bound for the total variation error of the distribution of the sample generated by the diffusion model under a mere sub-Gaussian assumption. If in addition, $p_0$ belongs to the nonparametric family of the $β$-Sobolev space with $β\le 2$, by adopting an early stopping strategy, we obtain that the diffusion model is nearly (up to log factors) minimax optimal. This removes the crucial lower bound assumption on $p_0$ in previous proofs of the minimax optimality of the diffusion model for nonparametric families.

math.ST↗

arXiv2024

Leveraging Self-Supervised Learning for MIMO-OFDM Channel Representation and Generation

In communications theory, the capacity of multiple input multiple output-orthogonal frequency division multiplexing (MIMO-OFDM) systems is fundamentally determined by wireless channels, which exhibit both diversity and correlation in spatial, frequency and temporal domains. It is further envisioned to exploit the inherent nature of channels, namely representation, to achieve geolocation-based MIMO transmission for 6G, exemplified by the fully-decoupled radio access network (FD-RAN). Accordingly, this paper first employs self-supervised learning to obtain channel representation from unlabeled channel, then proposes a channel generation assisted approach for determining MIMO precoding matrix solely based on geolocation. Specifically, we exploit the small-scale temporal domain variations of channels at a fixed geolocation, and design an ingenious pretext task tailored for contrastive learning. Then, a Transformer-based encoder is trained to output channel representations. We further develop a conditional diffusion generator to generate channel representations from geolocation. Finally, a Transformer-encoder-based decoder is utilized to reconstruct channels from generated representations, where the optimal channel is selected for calculating the precoding matrix for both single and dual BS transmission. We conduct experiments on a public ray-tracing channel dataset, and the extensive simulation results demonstrate the effectiveness of our channel representation method, and also showcase the performance improvement in geolocation-based MIMO transmission.

cs.IT↗

arXiv2024

Multivariate Priors and the Linearity of Optimal Bayesian Estimators under Gaussian Noise

Consider the task of estimating a random vector $X$ from noisy observations $Y = X + Z$, where $Z$ is a standard normal vector, under the $L^p$ fidelity criterion. This work establishes that, for $1 \leq p \leq 2$, the optimal Bayesian estimator is linear and positive definite if and only if the prior distribution on $X$ is a (non-degenerate) multivariate Gaussian. Furthermore, for $p > 2$, it is demonstrated that there are infinitely many priors that can induce such an estimator.

math.ST↗

arXiv2023

Enabling Feedback-Free MIMO Transmission for FD-RAN: A Data-driven Approach

To enhance flexibility and facilitate resource cooperation, a novel fully-decoupled radio access network (FD-RAN) architecture is proposed for 6G. However, the decoupling of uplink (UL) and downlink (DL) in FD-RAN makes the existing feedback mechanism ineffective. To this end, we propose an end-to-end data-driven MIMO solution without the conventional channel feedback procedure. Data-driven MIMO can alleviate the drawbacks of feedback including overheads and delay, and can provide customized precoding design for different BSs based on their historical channel data. It essentially learns a mapping from geolocation to MIMO transmission parameters. We first present a codebook-based approach, which selects transmission parameters from the statistics of discrete channel state information (CSI) values and utilizes integer interpolation for spatial inference. We further present a non-codebook-based approach, which 1) derives the optimal precoder from the singular value decomposition (SVD) of the channel; 2) utilizes variational autoencoder (VAE) to select the representative precoder from the latent Gaussian representations; and 3) exploits Gaussian process regression (GPR) to predict unknown precoders in the space domain. Extensive simulations are performed on a link-level 5G simulator using realistic ray-tracing channel data. The results demonstrate the effectiveness of data-driven MIMO, showcasing its potential for application in FD-RAN and 6G.

cs.NI↗

arXiv2023

An algorithm for $g$-invariant on unary Hermitian lattices over imaginary quadratic fields

Let $E=\mathbb{Q}\big(\sqrt{-d}\big)$ be an imaginary quadratic field for a square-free positive integer $d$, and let $\mathcal{O}$ be its ring of integers. For each positive integer $m$, let $I_m$ be the free Hermitian lattice over $\mathcal{O}$ with an orthonormal basis, let $\mathfrak{S}_d(1)$ be the set consisting of all positive definite integral unary Hermitian lattices over $\mathcal{O}$ that can be represented by some $I_m$, and let $g_d(1)$ be the least positive integer such that all Hermitian lattices in $\mathfrak{S}_d(1)$ can be uniformly represented by $I_{g_d(1)}$. The main results of this work provide an algorithm to calculate the explicit form of $\mathfrak{S}_d(1)$ and the exact value of $g_d(1)$ for every imaginary quadratic field $E$, which can be viewed as a natural extension of the Pythagoras number in the lattice setting.

math.NT↗

arXiv2023

A Few Interactions Improve Distributed Nonparametric Estimation, Optimally

Consider the problem of nonparametric estimation of an unknown $β$-Hölder smooth density $p_{XY}$ at a given point, where $X$ and $Y$ are both $d$ dimensional. An infinite sequence of i.i.d.\ samples $(X_i,Y_i)$ are generated according to this distribution, and two terminals observe $(X_i)$ and $(Y_i)$, respectively. They are allowed to exchange $k$ bits either in oneway or interactively in order for Bob to estimate the unknown density. We show that the minimax mean square risk is order $\left(\frac{k}{\log k} \right)^{-\frac{2β}{d+2β}}$ for one-way protocols and $k^{-\frac{2β}{d+2β}}$ for interactive protocols. The logarithmic improvement is nonexistent in the parametric counterparts, and therefore can be regarded as a consequence of nonparametric nature of the problem. Moreover, a few rounds of interactions achieve the interactive minimax rate: the number of rounds can grow as slowly as the super-logarithm (i.e., inverse tetration) of $k$. The proof of the upper bound is based on a novel multi-round scheme for estimating the joint distribution of a pair of biased Bernoulli variables, and the lower bound is built on a sharp estimate of a symmetric strong data processing constant for biased Bernoulli variables.

cs.IT↗

arXiv2023

Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling

Recently, Gilmer proved the first constant lower bound for the union-closed sets conjecture via an information-theoretic argument. The heart of the argument is an entropic inequality involving the OR function of two i.i.d.\ binary vectors, and the best constant obtainable through the i.i.d.\ coupling is $\frac{3-\sqrt{5}}{2}\approx0.38197$. Sawin demonstrated that the bound can be strictly improved by considering a convex combination of the i.i.d.\ coupling and the max-entropy coupling, and the best constant obtainable through this approach is around 0.38234, as evaluated by Yu and Cambie. In this work we show analytically that the bound can be further strictly improved by considering another class of coupling under which the two binary sequences are i.i.d.\ conditioned on an auxiliary random variable. We also provide a new class of bounds in terms of finite-dimensional optimization. For a basic instance from this class, analysis assisted with numerically solved 9-dimensional optimization suggests that the optimizer assumes a certain structure. Under numerically verified hypotheses, the lower bound for the union-closed sets conjecture can be improved to approximately 0.38271, a number that can be defined as the solution to an analytic equation.

cs.IT↗

arXiv2023

From Soft-Minoration to Information-Constrained Optimal Transport and Spiked Tensor Models

Let $P_Z$ be a given distribution on $\mathbb{R}^n$. For any $y\in\mathbb{R}^n$, we may interpret $ρ(y):=\ln\mathbb{E}[e^{\left }]$ as a soft-max of $\left $. We explore lower bounds on $\mathbb{E}[ρ(Y)]$ in terms of the minimum mutual information $I(Z,\bar{Z})$ over $P_{Z\bar{Z}}$ which is a coupling of $P_Z$ and itself such that $Z-\bar{Z}$ is bounded in a certain sense. This may be viewed as a soft version of Sudakov's minoration, which lower bounds the expected supremum of a stochastic process in terms of the packing number. Our method is based on convex geometry (thrifty approximation of convex bodies), and works for general non-Gaussian $Y$. When $Y$ is Gaussian and $\bar{Z}$ converges to $Z$, this recovers a recent inequality of Bai-Wu-Ozgur on information-constrained optimal transport, previously established using Gaussian-specific techniques. We also use soft-minoration to obtain asymptotically (in tensor order) tight bounds on the free energy in the Sherrington-Kirkpatrick model with spins uniformly distributed on a type class, implying asymptotically tight bounds for the type~II error exponent in spiked tensor detection.

cs.IT↗

arXiv2023

NeILF++: Inter-Reflectable Light Fields for Geometry and Material Estimation

We present a novel differentiable rendering framework for joint geometry, material, and lighting estimation from multi-view images. In contrast to previous methods which assume a simplified environment map or co-located flashlights, in this work, we formulate the lighting of a static scene as one neural incident light field (NeILF) and one outgoing neural radiance field (NeRF). The key insight of the proposed method is the union of the incident and outgoing light fields through physically-based rendering and inter-reflections between surfaces, making it possible to disentangle the scene geometry, material, and lighting from image observations in a physically-based manner. The proposed incident light and inter-reflection framework can be easily applied to other NeRF systems. We show that our method can not only decompose the outgoing radiance into incident lights and surface materials, but also serve as a surface refinement module that further improves the reconstruction detail of the neural surface. We demonstrate on several datasets that the proposed method is able to achieve state-of-the-art results in terms of geometry reconstruction quality, material estimation accuracy, and the fidelity of novel view rendering.

cs.CV↗

arXiv2022

$g$-invariant on unary Hermitian lattices over imaginary quadratic fields with class number $2$ or $3$

In this paper, we study the unary Hermitian lattices over imaginary quadratic fields. Let $E=\mathbb{Q}\big(\sqrt{-d}\big)$ be an imaginary quadratic field for a square-free positive integer $d$, and let $\mathcal{O}$ be its ring of integers. For each positive integer $m$, let $I_m$ be the free Hermitian lattice over $\mathcal{O}$ with an orthonormal basis, let $\mathfrak{S}_d(1)$ be the set consisting of all positive definite integral unary Hermitian lattices over $\mathcal{O}$ that can be represented by some $I_m$, and let $g_d(1)$ be the smallest positive integer such that all Hermitian lattices in $\mathfrak{S}_d(1)$ can be represented by $I_{g_d(1)}$ uniformly. The main results of this paper determine the explicit form of $\mathfrak{S}_d(1)$ and the exact value of $g_d(1)$ for every imaginary quadratic field $E$ with class number $2$ or $3$.

math.NT↗

arXiv2022

Stability of the Gaussian Stationary Point in the Han-Kobayashi Region for Z-Interference Channels

The Gaussian stationary point in an inequality motivated by the Z-interference channel was recently conjectured by Costa, Nair, Ng, and Wang to be the global optimizer, which, if true, would imply the optimality of the Han-Kobayashi region for the Gaussian Z-interference channel. This conjecture was known to be true for some parameter regimes, but the validity for all parameters, although suggested by Gaussian tensorization, was previously open. In this paper we construct several counterexamples showing that this conjecture may fail in certain regimes: A simple construction without Hermite polynomial perturbation is proposed, where distributions far from Gaussian are analytically shown to be better than the Gaussian stationary point. As alternatives, we consider perturbation along geodesics under either the $L^2$ or Wasserstein-2 metric, showing that the Gaussian stationary point is unstable in a certain regime. Similarity to stability of the Levy-Cramer theorem is discussed. The stability phase transition point admits a simple characterization in terms of the maximum eigenvalue of the Gaussian maximizer. Similar to the Holley-Stroock principle, we can show that in the stable regime the Gaussian stationary point is optimal in a neighborhood under the $L^{\infty}$-norm with respect to the Gaussian measure. For protocols with constant power control, our counterexamples imply Gaussian suboptimality for the Han-Kobayashi region. Allowing variable power control, we show that the Gaussian optimizers for the Han-Kobayashi region always lie in the stable regime. We propose an amended conjecture, whose validity would imply Gaussian optimality of the Han-Kobayashi bound in a certain regime.

cs.IT↗

arXiv2022

A new proof of Legendre's theorem on the Diophantine equation $ax^2+by^2+cz^2=0$

One of Legendre's theorems on the Diophantine equation $ax^2+by^2+cz^2=0$ provides necessary and sufficient conditions on the existence of nonzero rational solutions of this equation, which helps determine the existence of rational points on a conic. In this paper, we provide a new proof of this famous theorem using Hasse invariant and Jacobi symbol from the theory of quadratic forms.

math.NT↗

arXiv2022

NeILF: Neural Incident Light Field for Physically-based Material Estimation

We present a differentiable rendering framework for material and lighting estimation from multi-view images and a reconstructed geometry. In the framework, we represent scene lightings as the Neural Incident Light Field (NeILF) and material properties as the surface BRDF modelled by multi-layer perceptrons. Compared with recent approaches that approximate scene lightings as the 2D environment map, NeILF is a fully 5D light field that is capable of modelling illuminations of any static scenes. In addition, occlusions and indirect lights can be handled naturally by the NeILF representation without requiring multiple bounces of ray tracing, making it possible to estimate material properties even for scenes with complex lightings and geometries. We also propose a smoothness regularization and a Lambertian assumption to reduce the material-lighting ambiguity during the optimization. Our method strictly follows the physically-based rendering equation, and jointly optimizes material and lighting through the differentiable rendering process. We have intensively evaluated the proposed method on our in-house synthetic dataset, the DTU MVS dataset, and real-world BlendedMVS scenes. Our method is able to outperform previous methods by a significant margin in terms of novel view rendering quality, setting a new state-of-the-art for image-based material and lighting estimation.

cs.CV↗

arXiv2022

Minoration via Mixed Volumes and Cover's Problem for General Channels

We give a complete solution to an open problem of Thomas Cover in 1987 about the capacity of a relay channel in the general discrete memoryless setting without any additional assumptions. The key step in our approach is to lower bound a certain soft-max of a stochastic process by convex geometry methods, which is based on two ideas: First, the soft-max is lower bounded in terms of the supremum of another process, by approximating a convex set with a polytope with bounded number of vertices. Second, using a result of Pajor, the supremum of the process is lower bounded in terms of packing numbers by means of mixed-volume inequalities (Minkowski's first inequality).

cs.IT↗

arXiv2020

A Statistical Perspective on Coreset Density Estimation

Coresets have emerged as a powerful tool to summarize data by selecting a small subset of the original observations while retaining most of its information. This approach has led to significant computational speedups but the performance of statistical procedures run on coresets is largely unexplored. In this work, we develop a statistical framework to study coresets and focus on the canonical task of nonparameteric density estimation. Our contributions are twofold. First, we establish the minimax rate of estimation achievable by coreset-based estimators. Second, we show that the practical coreset kernel density estimators are near-minimax optimal over a large class of Hölder-smooth densities.

math.ST↗

arXiv2020

Efficient Interpolation of Density Estimators

We study the problem of space and time efficient evaluation of a nonparametric estimator that approximates an unknown density. In the regime where consistent estimation is possible, we use a piecewise multivariate polynomial interpolation scheme to give a computationally efficient construction that converts the original estimator to a new estimator that can be queried efficiently and has low space requirements, all without adversely deteriorating the original approximation quality. Our result gives a new statistical perspective on the problem of fast evaluation of kernel density estimators in the presence of underlying smoothness. As a corollary, we give a succinct derivation of a classical result of Kolmogorov---Tikhomirov on the metric entropy of Hölder classes of smooth functions.

math.ST↗

arXiv2020

A Note on Affine Invariant Cost Functions

We show that any affine invariant function on the set of positive definite matrices must factor through the determinant function, as long as the restriction of the function to scalar matrices is surjective. A motivation from robust statistics is discussed.

math.GR↗

arXiv2020

Power analysis of knockoff filters for correlated designs

The knockoff filter introduced by Barber and Candès 2016 is an elegant framework for controlling the false discovery rate in variable selection. While empirical results indicate that this methodology is not too conservative, there is no conclusive theoretical result on its power. When the predictors are i.i.d. Gaussian, it is known that as the signal to noise ratio tend to infinity, the knockoff filter is consistent in the sense that one can make FDR go to 0 and power go to 1 simultaneously. In this work we study the case where the predictors have a general covariance matrix $Σ$. We introduce a simple functional called effective signal deficiency (ESD) of the covariance matrix $Σ$ that predicts consistency of various variable selection methods. In particular, ESD reveals that the structure of the precision matrix $Σ^{-1}$ plays a central role in consistency and therefore, so does the conditional independence structure of the predictors. To leverage this connection, we introduce Conditional Independence knockoff, a simple procedure that is able to compete with the more sophisticated knockoff filters and that is defined when the predictors obey a Gaussian tree graphical models (or when the graph is sufficiently sparse). Our theoretical results are supported by numerical evidence on synthetic data.

math.ST↗