Learning and Testing Variable Partitions

$$Let $F$ be a multivariate function from a product set $\Sigma^n$ to an Abelian group $G$. A $k$-partition of $F$ with cost $\delta$ is a partition of the set of variables $\mathbf{V}$ into $k$ non-empty subsets $(\mathbf{X}_1, \dots, \mathbf{X}_k)$ such that $F(\mathbf{V})$ is $\delta$-close to $F_1(\mathbf{X}_1)+\dots+F_k(\mathbf{X}_k)$ for some $F_1, \dots, F_k$ with respect to a given error metric. We study algorithms for agnostically learning $k$ partitions and testing $k$-partitionability over various groups and error metrics given query access to $F$. In particular we show that $1.$ Given a function that has a $k$-partition of cost $\delta$, a partition of cost $\mathcal{O}(k n^2)(\delta + \epsilon)$ can be learned in time $\tilde{\mathcal{O}}(n^2 \mathrm{poly} (1/\epsilon))$ for any $\epsilon > 0$. In contrast, for $k = 2$ and $n = 3$ learning a partition of cost $\delta + \epsilon$ is NP-hard. $2.$ When $F$ is real-valued and the error metric is the 2-norm, a 2-partition of cost $\sqrt{\delta^2 + \epsilon}$ can be learned in time $\tilde{\mathcal{O}}(n^5/\epsilon^2)$. $3.$ When $F$ is $\mathbb{Z}_q$-valued and the error metric is Hamming weight, $k$-partitionability is testable with one-sided error and $\mathcal{O}(kn^3/\epsilon)$ non-adaptive queries. We also show that even two-sided testers require $\Omega(n)$ queries when $k = 2$. This work was motivated by reinforcement learning control tasks in which the set of control variables can be partitioned. The partitioning reduces the task into multiple lower-dimensional ones that are relatively easier to learn. Our second algorithm empirically increases the scores attained over previous heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202

Knowledge Extraction in Video Through the Interaction Analysis of Activities

Video is a massive amount of data that contains complex interactions between moving objects. The extraction of knowledge from this type of information creates a demand for video analytics systems that uncover statistical relationships between activities and learn the correspondence between content and labels. However, those are open research problems that have high complexity when multiple actors simultaneously perform activities, videos contain noise, and streaming scenarios are considered. The techniques introduced in this dissertation provide a basis for analyzing video. The primary contributions of this research consist of providing new algorithms for the efficient search of activities in video, scene understanding based on interactions between activities, and the predicting of labels for new scenes

SNIF TOOL - Sniffing for Patterns in Continuous Streams

Recent technological advances in sensor networks and mobile devices give rise to new challenges in processing of live streams. In particular, time-series sequence matching, namely, the similarity matching of live streams against a set of predefined pattern sequence queries, is an important technology for a broad range of domains that include monitoring the spread of hazardous waste and administering network traffic. In this thesis, I use the time critical application of monitoring of fire growth in an intelligent building as my motivating example. Various measures and algorithms have been established in the current literature for similarity of static time-series data. Matching continuous data poses the following new challenges: 1) fluctuations in stream characteristics, 2) real-time requirements of the application, 3) limited system resources, and, 4) noisy data. Thus the matching techniques proposed for static time-series are mostly not applicable for live stream matching. In this thesis, I propose a new generic framework, henceforth referred to as the n-Snippet Indices Framework (in short, SNIF), for discovering the similarity between a live stream and pattern sequences. The framework is composed of two key phases: (1.) Off-line preprocessing phase: where the pattern sequences are processed offline and stored into an approximate 2-level index structure; and (2.) On-line live stream matching phase: streaming time-series (or the live stream) is on-the-fly matched against the indexed pattern sequences. I introduce the concept of n-Snippets for numeric data as the unit for matching. The insight is to match small snippets of the live stream against prefixes of the patterns and maintain them in succession. Longer the pattern prefixes identified to be similar to the live stream, better the confirmation of the match. Thus, the live stream matching is performed in two levels of matching: bag matching for matching snippets and order checking for maintaining the lengths of the match. I propose four variations of matching algorithms that allow the user the capability to choose between the two conflicting characteristics of result accuracy versus response time. The effectiveness of SNIF to detect patterns has been thoroughly tested through extensive experimental evaluations using the continuous query engine CAPE as platform. The evaluations made use of real datasets from multiple domains, including fire monitoring, chlorine monitoring and sensor networks. Moreover, SNIF is demonstrated to be tolerant to noisy datasets

Doctor of Philosophy

dissertationKernel smoothing provides a simple way of finding structures in data sets without the imposition of a parametric model, for example, nonparametric regression and density estimates. However, in many data-intensive applications, the data set could be large. Thus, evaluating a kernel density estimate or kernel regression over the data set directly can be prohibitively expensive in big data. This dissertation is working on how to efficiently find a smaller data set that can approximate the original data set with a theoretical guarantee in the kernel smoothing setting and how to extend it to more general smooth range spaces. For kernel density estimates, we propose randomized and deterministic algorithms with quality guarantees that are orders of magnitude more efficient than previous algorithms, which do not require knowledge of the kernel or its bandwidth parameter and are easily parallelizable. Our algorithms are applicable to any large-scale data processing framework. We then further investigate how to measure the error between two kernel density estimates, which is usually measured either in L1 or L2 error. In this dissertation, we investigate the challenges in using a stronger error, L ∞ (or worst case) error. We present efficient solutions for how to estimate the L∞ error and how to choose the bandwidth parameter for a kernel density estimate built on a subsample of a large data set. We next extend smoothed versions of geometric range spaces from kernel range spaces to more general types of ranges, so that an element of the ground set can be contained in a range with a non-binary value in [0,1]. We investigate the approximation of these range spaces through ϵ-nets and ϵ-samples. Finally, we study coresets algorithms for kernel regression. The size of the coresets are independent of the size of the data set, rather they only depend on the error guarantee, and in some cases the size of domain and amount of smoothing. We evaluate our methods on very large time series and spatial data, demonstrate that they can be constructed extremely efficiently, and allow for great computational gains

GPU-based proximity query processing on unstructured triangular mesh model

This paper presents a novel proximity query (PQ) approach capable to detect the collision and calculate the minimal Euclidean distance between two non-convex objects in 3D, namely the robot and the environment. Such approaches are often considered as computationally demanding problems, but are of importance to many applications such as online simulation of haptic feedback and robot collision-free trajectory. Our approach enables to preserve the representation of unstructured environment in the form of triangular meshes. The proposed PQ algorithm is computationally parallel so that it can be effectively implemented on graphics processing units (GPUs). A GPU-based computation scheme is also developed and customized, which shows >200 times faster than an optimized CPU with single core. Comprehensive validation is also conducted on two simulated scenarios in order to demonstrate the practical values of its potential application in image-guided surgical robotics and humanoid robotic control.published_or_final_versio

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum