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 $\beta$-Sobolev space with $\beta\le 2$, 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.

$L^1$ Estimation: On the Optimality of Linear Estimators

Consider the problem of estimating a random variable $X$ from noisy observations $Y = X+ Z$, where $Z$ is standard normal, under the $L^1$ fidelity criterion. It is well known that the optimal Bayesian estimator in this setting is the conditional median. This work shows that the only prior distribution on $X$ that induces linearity in the conditional median is Gaussian. Along the way, several other results are presented. In particular, it is demonstrated that if the conditional distribution $P_{X|Y=y}$ is symmetric for all $y$, then $X$ must follow a Gaussian distribution. Additionally, we consider other $L^p$ losses and observe the following phenomenon: for $p \in [1,2]$, Gaussian is the only prior distribution that induces a linear optimal Bayesian estimator, and for $p \in (2,\infty)$, infinitely many prior distributions on $X$ can induce linearity. Finally, extensions are provided to encompass noise models leading to conditional distributions from certain exponential families.

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.

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.

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 H\"{o}lder classes of smooth functions.

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 H\"{o}lder-smooth densities.

Power analysis of knockoff filters for correlated designs

The knockoff filter introduced by Barber and Cand\`es 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 $\Sigma$. We introduce a simple functional called effective signal deficiency (ESD) of the covariance matrix $\Sigma$ that predicts consistency of various variable selection methods. In particular, ESD reveals that the structure of the precision matrix $\Sigma^{-1}$ 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.

Accuracy-Memory Tradeoffs and Phase Transitions in Belief Propagation

The analysis of Belief Propagation and other algorithms for the {\em reconstruction problem} plays a key role in the analysis of community detection in inference on graphs, phylogenetic reconstruction in bioinformatics, and the cavity method in statistical physics. We prove a conjecture of Evans, Kenyon, Peres, and Schulman (2000) which states that any bounded memory message passing algorithm is statistically much weaker than Belief Propagation for the reconstruction problem. More formally, any recursive algorithm with bounded memory for the reconstruction problem on the trees with the binary symmetric channel has a phase transition strictly below the Belief Propagation threshold, also known as the Kesten-Stigum bound. The proof combines in novel fashion tools from recursive reconstruction, information theory, and optimal transport, and also establishes an asymptotic normality result for BP and other message-passing algorithms near the critical threshold.

Communication Complexity of Estimating Correlations

We characterize the communication complexity of the following distributed estimation problem. Alice and Bob observe infinitely many iid copies of $\rho$-correlated unit-variance (Gaussian or $\pm1$ binary) random variables, with unknown $\rho\in[-1,1]$. By interactively exchanging $k$ bits, Bob wants to produce an estimate $\hat\rho$ of $\rho$. We show that the best possible performance (optimized over interaction protocol $\Pi$ and estimator $\hat \rho$) satisfies $\inf_{\Pi,\hat\rho}\sup_\rho \mathbb{E} [|\rho-\hat\rho|^2] = \Theta(\tfrac{1}{k})$. Furthermore, we show that the best possible unbiased estimator achieves performance of $1+o(1)\over {2k\ln 2}$. Curiously, thus, restricting communication to $k$ bits results in (order-wise) similar minimax estimation error as restricting to $k$ samples. Our results also imply an $\Omega(n)$ lower bound on the information complexity of the Gap-Hamming problem, for which we show a direct information-theoretic proof. Notably, the protocol achieving (almost) optimal performance is one-way (non-interactive). For one-way protocols we also prove the $\Omega(\tfrac{1}{k})$ bound even when $\rho$ is restricted to any small open sub-interval of $[-1,1]$ (i.e. a local minimax lower bound). %We do not know if this local behavior remains true in the interactive setting. Our proof techniques rely on symmetric strong data-processing inequalities, various tensorization techniques from information-theoretic interactive common-randomness extraction, and (for the local lower bound) on the Otto-Villani estimate for the Wasserstein-continuity of trajectories of the Ornstein-Uhlenbeck semigroup.