We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

As companies strive to develop artefacts intended for services instead of traditional sell-off, new challenges in the product development process arise to promote continuous improvement and increasing market profits. This creates a focus on product life-cycle components as companies then make life-cycle commitments, where they are responsible for the function availability during the extent of the life-cycle, i.e. functional products. One of these life-cycle components is manufacturing; therefore, companies search for new approaches of success during manufacturability evaluation already in engineering design. Efforts have been done to support early engineering design, as this phase sets constraints and opportunities for manufacturing. These efforts have turned into design for manufacturing methods and guidelines. A further step to improve the life-cycle focus during early engineering design is to reuse results and use experience from earlier projects. However, because results and experiences created during project work are often not documented for reuse, only remembered by some people, there is a need for design support. Knowledge engineering (KE) is a methodology for creating knowledge-based systems, e.g. systems that enable reuse of earlier results and make available both explicit and tacit corporate knowledge, enabling the automated generation and evaluation of new engineering design solutions during early product development. There are a variety of KE-approaches, such as knowledge-based engineering, case-based reasoning and programming, which have been used in research to develop design for manufacturing methods and applications. There are, however, opportunities for research where several approaches and their interdependencies, to create a transparent picture of how KE can be used to support engineering design, are investigated. The aim of the research presented in this thesis is to create new methods for design for manufacturing, by using several approaches of KE, and find the beneficial and less beneficial aspects of these methods in comparison to each other and earlier research. This thesis presents methods and applications for design for manufacturing using KE. KE has been employed in several ways, namely rule-based, rule-, programmingand finite element analysis (FEA)-based, and ruleand plan-based, which are tested and compared with each other. Results show that KE can be used to generate information about manufacturing in several ways. The rule-based way is suitable for supporting life-cycle commitments, as engineering design and manufacturing can be integrated with maintenance and performance predictions during early engineering design, though limited to the firing of production rules. The rule-, programmingand FEA-based way can be used to integrate computer-aided design tools and virtual manufacturing for non-linear stress and displacement analysis. This way may also bridge the gap between engineering designers and computational experts, even though this way requires a larger effort to program than the rule-based. The ruleand planbased way can enable design for manufacturing in two fashions – based on earlier manufacturing plans and based on rules. Because earlier manufacturing plans, together with programming algorithms, can handle knowledge that may be more intricate to capture as rules, as opposed to the time demanding routine work that is often automated by means of rules, several opportunities for designing for manufacturing exist.

Methods and Applications

We construct a stream-cipher S whose implementation is secure even if a bounded amount of arbitrary (adversarially chosen) information on the internal state ofS is leaked during computation. This captures all possible side-channel attacks on S where the amount of information leaked in a given period is bounded, but overall can be arbitrary large. The only other assumption we make on the implementation of S is that only data that is accessed during computation leaks information. The stream-cipher S generates its output in chunks K1, K2, . . . and arbitrary but bounded information leakage is modeled by allowing the adversary to adaptively chose a function fl : {0,1}* rarr {0, 1}lambda before Kl is computed, she then gets fl(taul) where taul is the internal state ofS that is accessed during the computation of Kg. One notion of security we prove for S is that Kg is indistinguishable from random when given K1,..., K1-1,f1(tau1 ),..., fl-1(taul-1) and also the complete internal state of S after Kg has been computed (i.e. S is forward-secure). The construction is based on alternating extraction (used in the intrusion-resilient secret-sharing scheme from FOCS'07). We move this concept to the computational setting by proving a lemma that states that the output of any PRG has high HILLpseudoentropy (i.e. is indistinguishable from some distribution with high min-entropy) even if arbitrary information about the seed is leaked. The amount of leakage lambda that we can tolerate in each step depends on the strength of the underlying PRG, it is at least logarithmic, but can be as large as a constant fraction of the internal state of S if the PRG is exponentially hard.

/pdf/leakage-resilient-cryptography-15i0byrphr.pdf

Leakage-Resilient Cryptography

Network management requires accurate estimates of metrics for traffic engineering (e.g., heavy hitters), anomaly detection (e.g., entropy of source addresses), and security (e.g., DDoS detection). Obtaining accurate estimates given router CPU and memory constraints is a challenging problem. Existing approaches fall in one of two undesirable extremes: (1) low fidelity general-purpose approaches such as sampling, or (2) high fidelity but complex algorithms customized to specific application-level metrics. Ideally, a solution should be both general (i.e., supports many applications) and provide accuracy comparable to custom algorithms. This paper presents UnivMon, a framework for flow monitoring which leverages recent theoretical advances and demonstrates that it is possible to achieve both generality and high accuracy. UnivMon uses an application-agnostic data plane monitoring primitive; different (and possibly unforeseen) estimation algorithms run in the control plane, and use the statistics from the data plane to compute application-level metrics. We present a proof-of-concept implementation of UnivMon using P4 and develop simple coordination techniques to provide a ``one-big-switch'' abstraction for network-wide monitoring. We evaluate the effectiveness of UnivMon using a range of trace-driven evaluations and show that it offers comparable (and sometimes better) accuracy relative to custom sketching solutions.

https://dl.acm.org/doi/pdf/10.1145/2934872.2934906

One Sketch to Rule Them All: Rethinking Network Flow Monitoring with UnivMon

Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was two-fold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine and b) considering graph problems yields new insights into the complexity of stream computation. However, the techniques developed in this area are now finding applications in other areas including data structures for dynamic graphs, approximation algorithms, and distributed and parallel computation. We survey the state-of-the-art results; identify general techniques; and highlight some simple algorithms that illustrate basic ideas.

/pdf/graph-stream-algorithms-a-survey-1m1oq4kh5y.pdf

Graph stream algorithms: a survey

We introduce a new method of maintaining a (k, )-coreset for clustering M -estimators over insertiononly streams. Let (P,w) be a weighted set (where w : P → [0,∞) is the weight function) of points in a ρ-metric space (meaning a set X equipped with a positive-semidefinite symmetric function D such that D(x, z) ≤ ρ(D(x, y) + D(y, z)) for all x, y, z ∈ X ). For any set of points C, we define COST(P,w,C) = ∑ p∈P w(p) minc∈C D(p, c). A (k, )-coreset for (P,w) is a weighted set (Q, v) such that for every set C of k points, (1 − )COST(P,w,C) ≤ COST(Q, v, C) ≤ (1 + )COST(P,w,C). Essentially, the coreset (Q, v) can be used in place of (P,w) for all operations concerning the COST function. Coresets, as a method of data reduction, are used to solve fundamental problems in machine learning of streaming and distributed data. M -estimators are functions D(x, y) that can be written as ψ(d(x, y)) where (X , d) is a true metric (i.e. 1-metric) space. Special cases of M -estimators include the well-known k-median (ψ(x) = x) and k-means (ψ(x) = x2) functions. Our technique takes an existing offline construction for an M -estimator coreset and converts it into the streaming setting, where n data points arrive sequentially. To our knowledge, this is the first streaming construction for any M -estimator that does not rely on the merge-and-reduce tree. For example, our coreset for streaming metric k-means uses O( −2k log k logn) points of storage. The previous state-of-the-art required storing at least O( −2k log k log4 n) points. 2012 ACM Subject Classification Theory of computation → Streaming models; Theory of computation → Facility location and clustering; Information systems → Query optimization

Streaming Coreset Constructions for M-Estimators.

Given a stream of data, a typical approach in streaming algorithms is to design a sophisticated algorithm with small memory that computes a specific statistic over the streaming data. Usually, if one wants to compute a different statistic after the stream is gone, it is impossible. But what if we want to compute a different statistic after the fact? In this paper, we consider the following fascinating possibility: can we collect some small amount of specific data during the stream that is "universal," i.e., where we do not know anything about the statistics we will want to later compute, other than the guarantee that had we known the statistic ahead of time, it would have been possible to do so with small memory? In other words, is it possible to collect some data in small space during the stream, such that any other statistic that can be computed with comparable space can be computed after the fact? This is indeed what we introduce (and show) in this paper with matching upper and lower bounds: we show that it is possible to collect universal statistics of polylogarithmic size, and prove that these universal statistics allow us after the fact to compute all other statistics that are computable with similar amounts of memory. We show that this is indeed possible, both for the standard unbounded streaming model and the sliding window streaming model.

Universal Streaming

The sliding window model generalizes the standard streaming model and often performs better in applications where recent data is more important or more accurate than data that arrived prior to a certain time. We study the problem of approximating symmetric norms (a norm on $\mathbb {R}^n$ that is invariant under sign-flips and coordinate-wise permutations) in the sliding window model, where only the W most recent updates define the underlying frequency vector. Whereas standard norm estimation algorithms for sliding windows rely on the smooth histogram framework of Braverman and Ostrovsky (FOCS 2007), analyzing the smoothness of general symmetric norms seems to be a challenging obstacle. Instead, we observe that the symmetric norm streaming algorithm of Braverman et al. (STOC 2017) can be reduced to identifying and approximating the frequency of heavy-hitters in a number of substreams. We introduce a heavy-hitter algorithm that gives a $(1+\epsilon )$-approximation to each of the reported frequencies in the sliding window model, thus obtaining the first algorithm for general symmetric norm estimation in the sliding window model. Our algorithm is a universal sketch that simultaneously approximates all symmetric norms in a parametrizable class and also improves upon the smooth histogram framework for estimating $L_p$ norms, for a range of large p. Finally, we consider the problem of overconstrained linear regression problem in the case that loss function that is an Orlicz norm, a symmetric norm that can be interpreted as a scale-invariant version of M-estimators. We give the first sublinear space algorithms that produce $(1+\epsilon )$-approximate solutions to the linear regression problem for loss functions that are Orlicz norms in both the streaming and sliding window models.

Symmetric Norm Estimation and Regression on Sliding Windows

We explore clustering problems in the streaming sliding window model in both general metric spaces and Euclidean space. We present the first polylogarithmic space $O(1)$-approximation to the metric $k$-median and metric $k$-means problems in the sliding window model, answering the main open problem posed by Babcock, Datar, Motwani and O'Callaghan, which has remained unanswered for over a decade. Our algorithm uses $O(k^3 \log^6 n)$ space and $\operatorname{poly}(k, \log n)$ update time. This is an exponential improvement on the space required by the technique due to Babcock, et al. We introduce a data structure that extends smooth histograms as introduced by Braverman and Ostrovsky to operate on a broader class of functions. In particular, we show that using only polylogarithmic space we can maintain a summary of the current window from which we can construct an $O(1)$-approximate clustering solution. 
Merge-and-reduce is a generic method in computational geometry for adapting offline algorithms to the insertion-only streaming model. Several well-known coreset constructions are maintainable in the insertion-only streaming model using this method, including well-known coreset techniques for the $k$-median, $k$-means in both low-and high-dimensional Euclidean spaces. Previous work has adapted these techniques to the insertion-deletion model, but translating them to the sliding window model has remained a challenge. We give the first algorithm that, given an insertion-only streaming coreset construction of space $s$, maintains a $(1\pm\epsilon)$-approximate coreset in the sliding window model using $O(s^2\epsilon^{-2}\log n)$ space. 
For clustering problems, our results constitute the first significant step towards resolving problem number 20 from the List of Open Problems in Sublinear Algorithms.

A Unified Approach for Clustering Problems on Sliding Windows

We analyze the popular kernel polynomial method (KPM) for approximating the spectral density (eigenvalue distribution) of an $n\times n$ Hermitian matrix $A$. We prove that a simple and practical variant of the KPM algorithm can approximate the spectral density to $\epsilon$ accuracy in the Wasserstein-1 distance with roughly $O({1}/{\epsilon})$ matrix-vector multiplications with $A$. This yields a provable linear time result for the problem with better $\epsilon$ dependence than prior work. 
The KPM variant we study is based on damped Chebyshev polynomial expansions. We show that it is stable, meaning that it can be combined with any approximate matrix-vector multiplication algorithm for $A$. As an application, we develop an $O(n\cdot \text{poly}(1/\epsilon))$ time algorithm for computing the spectral density of any $n\times n$ normalized graph adjacency or Laplacian matrix. This runtime is sublinear in the size of the matrix, and assumes sample access to the graph. 
Our approach leverages several tools from approximation theory, including Jackson's seminal work on approximation with positive kernels [Jackson, 1912], and stability properties of three-term recurrence relations for orthogonal polynomials.

Vladimir Braverman

Papers

Streaming Coreset Constructions for M-Estimators.

Universal Streaming

Symmetric Norm Estimation and Regression on Sliding Windows

A Unified Approach for Clustering Problems on Sliding Windows

Linear and Sublinear Time Spectral Density Estimation.