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

Approximating quantiles and distributions over streaming data has been studied for roughly two decades now. Recently, Karnin, Lang, and Liberty proposed the first asymptotically optimal algorithm for doing so. This manuscript complements their theoretical result by providing a practical variants of their algorithm with improved constants. For a given sketch size, our techniques provably reduce the upper bound on the sketch error by a factor of two. These improvements are verified experimentally. Our modified quantile sketch improves the latency as well by reducing the worst case update time from $O(1/\varepsilon)$ down to $O(\log (1/\varepsilon))$. We also suggest two algorithms for weighted item streams which offer improved asymptotic update times compared to naive extensions. Finally, we provide a specialized data structure for these sketches which reduces both their memory footprints and update times.

Streaming Quantiles Algorithms with Small Space and Update Time.

A central problem in data streams is to characterize which functions of an underlying frequency vector can be approximated efficiently. Recently there has been considerable effort in extending this problem to that of estimating functions of a matrix that is presented as a data-stream. This setting generalizes classical problems to the analogous ones for matrices. For example, instead of estimating frequent-item counts, we now wish to estimate "frequent-direction" counts. A related example is to estimate norms, which now correspond to estimating a vector norm on the singular values of the matrix. Despite recent efforts, the current understanding for such matrix problems is considerably weaker than that for vector problems. 
We study a number of aspects of estimating matrix norms in a stream that have not previously been considered: (1) multi-pass algorithms, (2) algorithms that see the underlying matrix one row at a time, and (3) time-efficient algorithms. Our multi-pass and row-order algorithms use less memory than what is provably required in the single-pass and entrywise-update models, and thus give separations between these models (in terms of memory). Moreover, all of our algorithms are considerably faster than previous ones. We also prove a number of lower bounds, and obtain for instance, a near-complete characterization of the memory required of row-order algorithms for estimating Schatten $p$-norms of sparse matrices.

/pdf/matrix-norms-in-data-streams-faster-multi-pass-and-row-order-2sy9nsring.pdf

Matrix Norms in Data Streams: Faster, Multi-Pass and Row-Order

Finding heavy-elements (heavy-hitters) in streaming data is one of the central, and well-understood tasks. Despite the importance of this problem, when considering the sliding windows model of streaming (where elements eventually expire) the problem of finding L 2-heavy elements has remained completely open despite multiple papers and considerable success in finding L 1-heavy elements.

How to Catch L 2 -Heavy-Hitters on Sliding Windows

Motivated by practical generalizations of the classic k-median and k-means objectives, such as clustering with size constraints, fair clustering, and Wasserstein barycenter, we introduce a meta-theorem for designing coresets for constrained-clustering problems. The meta-theorem reduces the task of coreset construction to one on a bounded number of ring instances with a much-relaxed additive error. This reduction enables us to construct coresets using uniform sampling, in contrast to the widely-used importance sampling, and consequently we can easily handle constrained objectives. Notably and perhaps surprisingly, this simpler sampling scheme can yield coresets whose size is independent of n, the number of input points. Our technique yields smaller coresets, and sometimes the first coresets, for a large number of constrained clustering problems, including capacitated clustering, fair clustering, Euclidean Wasserstein barycenter, clustering in minor-excluded graph, and polygon clustering under Fréchet and Hausdorff distance. Finally, our technique yields also smaller coresets for 1-median in low-dimensional Euclidean spaces, specifically of size $\tilde{O}(\varepsilon^{-15})$ in $\mathbb{R}^{2}$ and $\tilde{O}(\varepsilon^{-16})$ in $\mathbb{R}^{3}$.

The Power of Uniform Sampling for Coresets

Model compression is crucial for deployment of neural networks on devices with limited computational and memory resources. Many different methods show comparable accuracy of the compressed model and similar compression rates. However, the majority of the compression methods are based on heuristics and offer no worst-case guarantees on the trade-off between the compression rate and the approximation error for an arbitrarily new sample. We propose the first efficient structured pruning algorithm with a provable trade-off between its compression rate and the approximation error for any future test sample. Our method is based on the coreset framework and it approximates the output of a layer of neurons/filters by a coreset of neurons/filters in the previous layer and discards the rest. We apply this framework in a layer-by-layer fashion from the bottom to the top. Unlike previous works, our coreset is data independent, meaning that it provably guarantees the accuracy of the function for any input $x\in \mathbb{R}^d$, including an adversarial one.

Vladimir Braverman

Papers

Streaming Quantiles Algorithms with Small Space and Update Time.

Matrix Norms in Data Streams: Faster, Multi-Pass and Row-Order

How to Catch L 2 -Heavy-Hitters on Sliding Windows

The Power of Uniform Sampling for Coresets

Data-Independent Structured Pruning of Neural Networks via Coresets