Demystifying Fixed k-Nearest Neighbor Information Estimators

Estimating mutual information from i.i.d. samples drawn from an unknown joint density function is a basic statistical problem of broad interest with multitudinous applications. The most popular estimator is one proposed by Kraskov and St\"ogbauer and Grassberger (KSG) in 2004, and is nonparametric and based on the distances of each sample to its $k^{\rm th}$ nearest neighboring sample, where $k$ is a fixed small integer. Despite its widespread use (part of scientific software packages), theoretical properties of this estimator have been largely unexplored. In this paper we demonstrate that the estimator is consistent and also identify an upper bound on the rate of convergence of the bias as a function of number of samples. We argue that the superior performance benefits of the KSG estimator stems from a curious "correlation boosting" effect and build on this intuition to modify the KSG estimator in novel ways to construct a superior estimator. As a byproduct of our investigations, we obtain nearly tight rates of convergence of the $\ell_2$ error of the well known fixed $k$ nearest neighbor estimator of differential entropy by Kozachenko and Leonenko.Comment: 55 pages, 8 figure

On the bit error rate of repeated error-correcting codes

Classically, error-correcting codes are studied with respect to performance metrics such as minimum distance (combinatorial) or probability of bit/block error over a given stochastic channel. In this paper, a different metric is considered. It is assumed that the block code is used to repeatedly encode user data. The resulting stream is subject to adversarial noise of given power, and the decoder is required to reproduce the data with minimal possible bit-error rate. This setup may be viewed as a combinatorial joint source-channel coding. Two basic results are shown for the achievable noise-distortion tradeoff: the optimal performance for decoders that are informed of the noise power, and global bounds for decoders operating in complete oblivion (with respect to noise level). General results are applied to the Hamming [7, 4, 3] code, for which it is demonstrated (among other things) that no oblivious decoder exist that attains optimality for all noise levels simultaneously.National Science Foundation (U.S.) (Grant CCF-13-18620

Discovering Potential Correlations via Hypercontractivity

Discovering a correlation from one variable to another variable is of fundamental scientific and practical interest. While existing correlation measures are suitable for discovering average correlation, they fail to discover hidden or potential correlations. To bridge this gap, (i) we postulate a set of natural axioms that we expect a measure of potential correlation to satisfy; (ii) we show that the rate of information bottleneck, i.e., the hypercontractivity coefficient, satisfies all the proposed axioms; (iii) we provide a novel estimator to estimate the hypercontractivity coefficient from samples; and (iv) we provide numerical experiments demonstrating that this proposed estimator discovers potential correlations among various indicators of WHO datasets, is robust in discovering gene interactions from gene expression time series data, and is statistically more powerful than the estimators for other correlation measures in binary hypothesis testing of canonical examples of potential correlations.Comment: 30 pages, 19 figures, accepted for publication in the 31st Conference on Neural Information Processing Systems (NIPS 2017

Conditional dependence via Shannon capacity: axioms, estimators and applications

We consider axiomatically the problem of estimating the strength of a conditional dependence relationship P_{Y|X} from a random variable X to a random variable Y. This has applications in determining the strength of a known causal relationship, where the strength depends only on the conditional distribution of the effect given the cause (and not on the driving distribution of the cause). Shannon capacity, appropriately regularized, emerges as a natural measure under these axioms. We examine the problem of calculating Shannon capacity from the observed samples and propose a novel fixed-k nearest-neighbor estimator, and demonstrate its consistency. Finally, we demonstrate an application to single-cell flow-cytometry where the proposed estimators significantly reduce sample complexity

InstantMesh: Efficient 3D Mesh Generation from a Single Image with Sparse-view Large Reconstruction Models

We present InstantMesh, a feed-forward framework for instant 3D mesh generation from a single image, featuring state-of-the-art generation quality and significant training scalability. By synergizing the strengths of an off-the-shelf multiview diffusion model and a sparse-view reconstruction model based on the LRM architecture, InstantMesh is able to create diverse 3D assets within 10 seconds. To enhance the training efficiency and exploit more geometric supervisions, e.g, depths and normals, we integrate a differentiable iso-surface extraction module into our framework and directly optimize on the mesh representation. Experimental results on public datasets demonstrate that InstantMesh significantly outperforms other latest image-to-3D baselines, both qualitatively and quantitatively. We release all the code, weights, and demo of InstantMesh, with the intention that it can make substantial contributions to the community of 3D generative AI and empower both researchers and content creators.Comment: Technical report. Project: https://github.com/TencentARC/InstantMes

Dream3D: Zero-Shot Text-to-3D Synthesis Using 3D Shape Prior and Text-to-Image Diffusion Models

Recent CLIP-guided 3D optimization methods, such as DreamFields and PureCLIPNeRF, have achieved impressive results in zero-shot text-to-3D synthesis. However, due to scratch training and random initialization without prior knowledge, these methods often fail to generate accurate and faithful 3D structures that conform to the input text. In this paper, we make the first attempt to introduce explicit 3D shape priors into the CLIP-guided 3D optimization process. Specifically, we first generate a high-quality 3D shape from the input text in the text-to-shape stage as a 3D shape prior. We then use it as the initialization of a neural radiance field and optimize it with the full prompt. To address the challenging text-to-shape generation task, we present a simple yet effective approach that directly bridges the text and image modalities with a powerful text-to-image diffusion model. To narrow the style domain gap between the images synthesized by the text-to-image diffusion model and shape renderings used to train the image-to-shape generator, we further propose to jointly optimize a learnable text prompt and fine-tune the text-to-image diffusion model for rendering-style image generation. Our method, Dream3D, is capable of generating imaginative 3D content with superior visual quality and shape accuracy compared to state-of-the-art methods.Comment: Accepted by CVPR 2023. Project page: https://bluestyle97.github.io/dream3d