Fast Binary Embedding via Circulant Downsampled Matrix -- A Data-Independent Approach

循环Johnson-Lindenstrauss嵌入的新界

New bounds for circulant Johnson-Lindenstrauss embeddings

本文分析了循环Johnson-Lindenstrauss（JL）嵌入，它是一类重要的结构化随机JL嵌入，是通过随机化由随机向量生成的循环矩阵的列符号而形成的.利用最近的解耦技巧和矩阵值Bernstein不等式，我们得到了高斯循环JL嵌入的一个新的界$k=O(ε^{-2}\log^{(1+δ)} (n))$.利用拉普拉斯变换技巧（也称伯恩斯坦技巧），将结果推广到亚高斯情形。本文中的边界提供了一个小的改进，目前的最佳界限高斯循环JL嵌入某些参数制度，并使用更直接的方法。

Fast binary embeddings with Gaussian circulant matrices: improved bounds

We consider the problem of encoding a finite set of vectors into a small number of bits while approximately retaining information on the angular distances between the vectors. By deriving improved variance bounds related to binary Gaussian circulant embeddings, we largely fix a gap in the proof of the best known fast binary embedding method. Our bounds also show that well-spreadness assumptions on the data vectors, which were needed in earlier work on variance bounds, are unnecessary. In addition, we propose a new binary embedding with a faster running time on sparse data.

Numerically stable coded matrix computations via circulant and rotation matrix embeddings

Polynomial based methods have recently been used in several works for mitigating the effect of stragglers (slow or failed nodes) in distributed matrix computations. For a system with $n$ worker nodes where $s$ can be stragglers, these approaches allow for an optimal recovery threshold, whereby the intended result can be decoded as long as any $(n-s)$ worker nodes complete their tasks. However, they suffer from serious numerical issues owing to the condition number of the corresponding real Vandermonde-structured recovery matrices; this condition number grows exponentially in $n$. We present a …

Fast binary embeddings, and quantized compressed sensing with structured matrices

This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset $\mathcal{X} \subset \mathbb{R}^n$, associated with the Euclidean metric, with points in the cube $\{\pm 1\}^m$ and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in $\mathcal{X}$. Our methods rely on quantizing fast Johnson-Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our…

Fast metric embedding into the Hamming cube

We consider the problem of embedding a subset of $\mathbb{R}^n$ into a low-dimensional Hamming cube in an almost isometric way. We construct a simple, data-oblivious, and computationally efficient map that achieves this task with high probability: we first apply a specific structured random matrix, which we call the double circulant matrix; using that matrix requires linear storage and matrix-vector multiplication can be performed in near-linear time. We then binarize each vector by comparing each of its entries to a random threshold, selected uniformly at random from a well-chosen interval. W…

具有快速乘法和较少行的RIP矩阵的新构造

New constructions of RIP matrices with fast multiplication and fewer rows

在压缩感知中，“受限等距属性”（RIP）是从m个线性测量值Φ x有效重构C^d中的近k稀疏向量x的充分条件。期望m较小，并且Φ支持快速矩阵-向量乘法。本文给出了C^{mxd}中RIP矩阵Phi的一个随机构造，它保持了所有k-稀疏向量的L_2范数，其中矩阵与向量的乘积Phix可以在近似线性时间内计算.行数m大约为eps^{-2}klog dlog ^2（klog d）。先前对支持快速矩阵-向量乘法的RIP矩阵（例如采样离散傅立叶矩阵）的构造的分析要求m大大约一个log k因子。支持快速矩阵向量乘法对于重复乘以Φ或Φ ^* 的迭代恢复算法是有用的。此外，我们的结构，以及RIP矩阵和约翰逊-林登施特劳斯引理之间的联系[Krahmer-Ward，SIAM。J. Math. Anal. 2011年]，意味着快速的约翰逊-林登施特劳斯嵌入与渐近更少的行比以前已知的。我们的方法是对以前结构的简单扭曲。我们不是选择嵌入矩阵的行是从一些更大的结构化矩阵（如离散傅立叶变换或随机循环矩阵）中采样的行，而是选择嵌入矩阵的每行是原始矩阵的少量行的线性组合，并将随机符号翻转作为系数。在我们的分析中的主要工具是最近在[Krahmer-Mendelson-Rauhut，arXiv：1207.0235]中某些类型的Rademacher混沌过程的上确界。