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 present data streaming algorithms for the $k$-median problem in high-dimensional dynamic geometric data streams, i.e. streams allowing both insertions and deletions of points from a discrete Euclidean space $\{1, 2, \ldots \Delta\}^d$. Our algorithms use $k \epsilon^{-2} poly(d \log \Delta)$ space/time and maintain with high probability a small weighted set of points (a coreset) such that for every set of $k$ centers the cost of the coreset $(1+\epsilon)$-approximates the cost of the streamed point set. We also provide algorithms that guarantee only positive weights in the coreset with additional logarithmic factors in the space and time complexities. We can use this positively-weighted coreset to compute a $(1+\epsilon)$-approximation for the $k$-median problem by any efficient offline $k$-median algorithm. All previous algorithms for computing a $(1+\epsilon)$-approximation for the $k$-median problem over dynamic data streams required space and time exponential in $d$. Our algorithms can be generalized to metric spaces of bounded doubling dimension.

Clustering High Dimensional Dynamic Data Streams

We use differential equations based approaches to provide some {\it \textbf{physics}} insights into analyzing the dynamics of popular optimization algorithms in machine learning. In particular, we study gradient descent, proximal gradient descent, coordinate gradient descent, proximal coordinate gradient, and Newton's methods as well as their Nesterov's accelerated variants in a unified framework motivated by a natural connection of optimization algorithms to physical systems. Our analysis is applicable to more general algorithms and optimization problems {\it \textbf{beyond}} convexity and strong convexity, e.g. Polyak-\L ojasiewicz and error bound conditions (possibly nonconvex).

/pdf/the-physical-systems-behind-optimization-algorithms-70kx81j46x.pdf

The Physical Systems Behind Optimization Algorithms

We design new sketching algorithms for unitarily invariant matrix norms, including the Schatten $p$-norms~$\|{\cdot}\|_{S_p}$, and obtain, as a by-product, streaming algorithms that approximate the norm of a matrix $A$ presented as a turnstile data stream. The primary advantage of our streaming algorithms is that they are simpler and faster than previous algorithms, while requiring the same or less storage. Our three main results are a faster sketch for estimating $\|{A}\|_{S_p}$, a smaller-space $O(1)$-pass sketch for $\|{A}\|_{S_p}$, and more general sketching technique that yields sublinear-space approximations for a wide class of matrix norms. These improvements are powered by dimensionality reduction techniques that are modern incarnations of the Johnson-Lindenstrauss Lemma~\cite{JL84}. When $p\geq 2$ is even or $A$ is PSD, our fast one-pass algorithm approximates $\|{A}\|_{S_p}$ in optimal, $n^{2-4/p}$, space with $O(1)$ update time and $o(n^{2.4(1-2/p)})$ time to extract the approximation from the sketch, while the $\lceil p/2\rceil$-pass algorithm is built on a smaller sketch of size $n^{1-1/(p-1)}$ with $O(1)$ update time and $n^{1-1/(p-1)}$ query time. Finally, for a PSD matrix $A$ and a unitarily invariant norm $l(\cdot)$, we prove that one can obtain an approximation to $l(A)$ from a sketch $GAH^T$ where $G$ and $H$ are independent Oblivious Subspace Embeddings and the dimension of the sketch is polynomial in the intrinsic dimension of $A$. The intrinsic dimension of a matrix is a robust version of the rank that is equal to the ratio $\sum_i\sigma_i/\sigma_1$. It is small, e.g., for models in machine learning which consist of a low rank matrix plus noise. Naturally, this leads to much smaller sketches for many norms.

Sketches for Matrix Norms: Faster, Smaller and More General.

We initiate the study of numerical linear algebra in the sliding window model, where only the most recent $W$ updates in the data stream form the underlying set. Although most existing work in the sliding window model uses the smooth histogram framework, most interesting linear-algebraic problems are not smooth; we show that the spectral norm, vector induced matrix norms, generalized regression, and low-rank approximation are not amenable for the smooth histogram framework. To overcome this challenge, we first give a deterministic algorithm that achieves spectral approximation in the sliding window model that can be viewed as a generalization of smooth histograms, using the Loewner ordering of PSD matrices. We then give algorithms for both spectral approximation and low-rank approximation that are space-optimal up to polylogarithmic factors. Our algorithms are based on a new notion of "reverse online" leverage scores that account for both how unique and how recent a row is, while preserving sparsity so that both our algorithms run in input sparsity runtime, up to lower order factors. We show that our techniques have applications to linear-algebraic problems in other settings. Specifically, we show that our analysis immediately implies an algorithm for low-rank approximation in the online setting that is space-optimal up to logarithmic factors, as well as nearly input sparsity time. We show our deterministic spectral approximation algorithm can be used to handle $\ell_1$ spectral approximation in the sliding window model under a certain assumption on the bit complexity of the entries. Finally, we show that our downsampling framework can be applied to the problem of approximate matrix multiplication and provide upper and lower bounds that are tight up to $\log\log W$ factors.

Numerical Linear Algebra in the Sliding Window Model

Spectral functions of large matrices contains important structural information about the underlying data, and is thus becoming increasingly important. Many times, large matrices representing real-world data are \emph{sparse} or \emph{doubly sparse} (i.e., sparse in both rows and columns), and are accessed as a \emph{stream} of updates, typically organized in \emph{row-order}. In this setting, where space (memory) is the limiting resource, all known algorithms require space that is polynomial in the dimension of the matrix, even for sparse matrices. We address this challenge by providing the first algorithms whose space requirement is \emph{independent of the matrix dimension}, assuming the matrix is doubly-sparse and presented in row-order. Our algorithms approximate the Schatten $p$-norms, which we use in turn to approximate other spectral functions, such as logarithm of the determinant, trace of matrix inverse, and Estrada index. We validate these theoretical performance bounds by numerical experiments on real-world matrices representing social networks. We further prove that multiple passes are unavoidable in this setting, and show extensions of our primary technique, including a trade-off between space requirements and number of passes.

Vladimir Braverman

Papers

Clustering High Dimensional Dynamic Data Streams

The Physical Systems Behind Optimization Algorithms

Sketches for Matrix Norms: Faster, Smaller and More General.

Numerical Linear Algebra in the Sliding Window Model

Schatten Norms in Matrix Streams: Hello Sparsity, Goodbye Dimension