There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Data Mining - Concepts and Techniques.

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.

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Power-Law Distributions in Empirical Data

Fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.

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Information Theory, Inference and Learning Algorithms

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

In the standard set reconciliation problem, there are two parties A 1 and A 2 , each respectively holding a set of elements S 1 and S 2 . The goal is for both parties to obtain the union S 1 U S 2 . In many distributed computing settings the sets may be large but the set difference |S 1 − S 2 | +|S 2 − S 1 | is small. In these cases one aims to achieve reconciliation efficiently in terms of communication; ideally, the communication should depend on the size of the set difference, and not on the size of the sets. Recent work has considered generalizations of the reconciliation problem to multi-party settings, using a framework based on a specific type of linear sketch called an Invertible Bloom Lookup Table. Here, we consider multi-party set reconciliation using the alternative framework of characteristic polynomials, which have previously been used for efficient pairwise set reconciliation protocols, and compare their performance with Invertible Bloom Lookup Tables for these problems.

Multi-party set reconciliation using characteristic polynomials

Programmable switches are driving a massive increase in fine-grained measurements. This puts significant pressure on telemetry collectors that have to process reports from many switches. Past research acknowledged this problem by either improving collectors' stack performance or by limiting the amount of data sent from switches. In this paper, we take a different and radical approach: switches are responsible for directly inserting queryable telemetry data into the collectors' memory, bypassing their CPU, and thereby improving their collection scalability. We propose to use a method we call direct telemetry access, where switches jointly write telemetry reports directly into the same collector's memory region, without coordination. Our solution, DART, is probabilistic, trading memory redundancy and query success probability for CPU resources at collectors. We prototype DART using commodity hardware such as P4 switches and RDMA NICs and show that we get high query success rates with a reasonable memory overhead. For example, we can collect INT path tracing information on a fat tree topology without a collector's CPU involvement while achieving 99.9% query success probability and using just 300 bytes per flow.

Zero-CPU Collection with Direct Telemetry Access

Search engines often employ techniques for determining syntactic similarity of Web pages. Such a tool allows them to avoid returning multiple copies of essentially the same page when a user makes a query. Here we describe our experience extending these techniques to MIDI music files. The music domain requires modification to cope with problems introduced in the musical setting, such as polyphony. Our experience suggests that when used properly these techniques prove useful for determining duplicates and clustering databases in the musical setting as well.

/pdf/estimating-resemblance-of-midi-documents-3egwo394hx.pdf

Estimating Resemblance of MIDI Documents

In this work we consider the problem of recovering $n$ discrete random variables $x_i\in \{0,\ldots,k-1\}, 1 \leq i \leq n$ (where $k$ is constant) with the smallest possible number of queries to a noisy oracle that returns for a given query pair $(x_i,x_j)$ a noisy measurement of their modulo $k$ pairwise difference, i.e., $y_{ij} = (x_i-x_j) \mod k$. This is a joint discrete alignment problem with important applications in computer vision, graph mining, and spectroscopy imaging. Our main result is a polynomial time algorithm that learns exactly with high probability the alignment (up to some unrecoverable offset) using $O(n^{1+o(1)})$ queries.

Joint Alignment From Pairwise Differences with a Noisy Oracle

We introduced the concept of a digital fountain as a scalable approach to reliable multicast, realized with fast and practical erasure codes, in a paper published in ACM SIGCOMM '98. This invited editorial, on the occasion of the 50th anniversary of the SIG, reflects on the trajectory of work leading up to our approach, and the numerous developments in the field in the subsequent 21 years. We discuss advances in rateless codes, efficient implementations, applications of digital fountains in distributed storage systems, and connections to invertible Bloom lookup tables.

Michael Mitzenmacher

Papers

Multi-party set reconciliation using characteristic polynomials

Zero-CPU Collection with Direct Telemetry Access

Estimating Resemblance of MIDI Documents

Joint Alignment From Pairwise Differences with a Noisy Oracle

A digital fountain retrospective