Fast Detection of Outliers in Data Streams with the $Q_n$ Estimator

The Optimal Quantile Estimator for Compressed Counting

Compressed Counting (CC) was recently proposed for very efficiently computing the (approximate) $α$th frequency moments of data streams, where $0<α<= 2$. Several estimators were reported including the geometric mean estimator, the harmonic mean estimator, the optimal power estimator, etc. The geometric mean estimator is particularly interesting for theoretical purposes. For example, when $α-> 1$, the complexity of CC (using the geometric mean estimator) is $O(1/ε)$, breaking the well-known large-deviation bound $O(1/ε^2)$. The case $α\approx 1$ has important applications, for example, computin…

重尾污染比对抗性污染更容易处理

Heavy-tailed Contamination is Easier than Adversarial Contamination

本文证明任何对抗性鲁棒估计器对重尾离群值同样有效，并构造反例说明重尾估计器无法直接用于对抗性设定，从而表明重尾估计比对抗性鲁棒估计更容易，为重尾场景开辟了新算法方向。

ExtendedHyperLogLog: Analysis of a new Cardinality Estimator

We discuss the problem of counting distinct elements in a stream. A stream is usually considered as a sequence of elements that come one at a time. An exact solution to the problem requires memory space of the size of the stream. For many applications this solution is infeasible due to very large streams. The solution in that case, is to use a probabilistic data structure (also called sketch), from which we can estimate with high accuracy the cardinality of the stream. We present a new algorithm, ExtendedHyperLogLog (EHLL), which is based on the state-of-the-art algorithm, HyperLogLog (HLL). I…

A Framework for Adversarial Streaming via Differential Privacy and Difference Estimators

Classical streaming algorithms operate under the (not always reasonable) assumption that the input stream is fixed in advance. Recently, there is a growing interest in designing robust streaming algorithms that provide provable guarantees even when the input stream is chosen adaptively as the execution progresses. We propose a new framework for robust streaming that combines techniques from two recently suggested frameworks by Hassidim et al. [NeurIPS 2020] and by Woodruff and Zhou [FOCS 2021]. These recently suggested frameworks rely on very different ideas, each with its own strengths and we…

Efficient Subspace Search in Data Streams

In the real world, data streams are ubiquitous -- think of network traffic or sensor data. Mining patterns, e.g., outliers or clusters, from such data must take place in real time. This is challenging because (1) streams often have high dimensionality, and (2) the data characteristics may change over time. Existing approaches tend to focus on only one aspect, either high dimensionality or the specifics of the streaming setting. For static data, a common approach to deal with high dimensionality -- known as subspace search -- extracts low-dimensional, `interesting' projections (subspaces), in w…

Algorithms and Hardness for Robust Subspace Recovery

We consider a fundamental problem in unsupervised learning called \emph{subspace recovery}: given a collection of $m$ points in $\mathbb{R}^n$, if many but not necessarily all of these points are contained in a $d$-dimensional subspace $T$ can we find it? The points contained in $T$ are called {\em inliers} and the remaining points are {\em outliers}. This problem has received considerable attention in computer science and in statistics. Yet efficient algorithms from computer science are not robust to {\em adversarial} outliers, and the estimators from robust statistics are hard to compute in …

Outlier-Robust High-Dimensional Sparse Estimation via Iterative Filtering

We study high-dimensional sparse estimation tasks in a robust setting where a constant fraction of the dataset is adversarially corrupted. Specifically, we focus on the fundamental problems of robust sparse mean estimation and robust sparse PCA. We give the first practically viable robust estimators for these problems. In more detail, our algorithms are sample and computationally efficient and achieve near-optimal robustness guarantees. In contrast to prior provable algorithms which relied on the ellipsoid method, our algorithms use spectral techniques to iteratively remove outliers from the d…

Outlier Robust Mean Estimation with Subgaussian Rates via Stability

We study the problem of outlier robust high-dimensional mean estimation under a finite covariance assumption, and more broadly under finite low-degree moment assumptions. We consider a standard stability condition from the recent robust statistics literature and prove that, except with exponentially small failure probability, there exists a large fraction of the inliers satisfying this condition. As a corollary, it follows that a number of recently developed algorithms for robust mean estimation, including iterative filtering and non-convex gradient descent, give optimal error estimators with …

Pattern Recognition and Event Detection on IoT Data-streams

Big data streams are possibly one of the most essential underlying notions. However, data streams are often challenging to handle owing to their rapid pace and limited information lifetime. It is difficult to collect and communicate stream samples while storing, transmitting and computing a function across the whole stream or even a large segment of it. In answer to this research issue, many streaming-specific solutions were developed. Stream techniques imply a limited capacity of one or more resources such as computing power and memory, as well as time or accuracy limits. Reservoir sampling a…