arXiv subjects

Jeff M. Phillips

Publications and source records attributed to Jeff M. Phillips.

At least 55 records · Page 3Linked to original sources

arXiv2021

A Deterministic Streaming Sketch for Ridge Regression

We provide a deterministic space-efficient algorithm for estimating ridge regression. For $n$ data points with $d$ features and a large enough regularization parameter, we provide a solution within $\varepsilon$ L$_2$ error using only $O(d/\varepsilon)$ space. This is the first $o(d^2)$ space deterministic streaming algorithm with guaranteed solution error and risk bound for this classic problem. The algorithm sketches the covariance matrix by variants of Frequent Directions, which implies it can operate in insertion-only streams and a variety of distributed data settings. In comparisons to randomized sketching algorithms on synthetic and real-world datasets, our algorithm has less empirical error using less space and similar time.

cs.LG↗

arXiv2020

Closed Form Word Embedding Alignment

We develop a family of techniques to align word embeddings which are derived from different source datasets or created using different mechanisms (e.g., GloVe or word2vec). Our methods are simple and have a closed form to optimally rotate, translate, and scale to minimize root mean squared errors or maximize the average cosine similarity between two embeddings of the same vocabulary into the same dimensional space. Our methods extend approaches known as Absolute Orientation, which are popular for aligning objects in three-dimensions, and generalize an approach by Smith etal (ICLR 2017). We prove new results for optimal scaling and for maximizing cosine similarity. Then we demonstrate how to evaluate the similarity of embeddings from different sources or mechanisms, and that certain properties like synonyms and analogies are preserved across the embeddings and can be enhanced by simply aligning and averaging ensembles of embeddings.

cs.CL↗

arXiv2020

The GaussianSketch for Almost Relative Error Kernel Distance

We introduce two versions of a new sketch for approximately embedding the Gaussian kernel into Euclidean inner product space. These work by truncating infinite expansions of the Gaussian kernel, and carefully invoking the RecursiveTensorSketch [Ahle et al. SODA 2020]. After providing concentration and approximation properties of these sketches, we use them to approximate the kernel distance between points sets. These sketches yield almost $(1+\varepsilon)$-relative error, but with a small additive $α$ term. In the first variants the dependence on $1/α$ is poly-logarithmic, but has higher degree of polynomial dependence on the original dimension $d$. In the second variant, the dependence on $1/α$ is still poly-logarithmic, but the dependence on $d$ is linear.

cs.LG↗

arXiv2019

Finding the Mode of a Kernel Density Estimate

Given points $p_1, \dots, p_n$ in $\mathbb{R}^d$, how do we find a point $x$ which maximizes $\frac{1}{n} \sum_{i=1}^n e^{-\|p_i - x\|^2}$? In other words, how do we find the maximizing point, or mode of a Gaussian kernel density estimation (KDE) centered at $p_1, \dots, p_n$? Given the power of KDEs in representing probability distributions and other continuous functions, the basic mode finding problem is widely applicable. However, it is poorly understood algorithmically. Few provable algorithms are known, so practitioners rely on heuristics like the "mean-shift" algorithm, which are not guaranteed to find a global optimum. We address this challenge by providing fast and provably accurate approximation algorithms for mode finding in both the low and high dimensional settings. For low dimension $d$, our main contribution is to reduce the mode finding problem to a solving a small number of systems of polynomial inequalities. For high dimension $d$, we prove the first dimensionality reduction result for KDE mode finding, which allows for reduction to the low dimensional case. Our result leverages Johnson-Lindenstrauss random projection, Kirszbraun's classic extension theorem, and perhaps surprisingly, the mean-shift heuristic for mode finding.

cs.DS↗

arXiv2019

The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

The Vapnik-Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set $X$ is a set of polygonal curves in $\mathbb{R}^d$ and the sets $\mathcal{R}$ are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

cs.CG↗

arXiv2019

The Kernel Spatial Scan Statistic

Kulldorff's (1997) seminal paper on spatial scan statistics (SSS) has led to many methods considering different regions of interest, different statistical models, and different approximations while also having numerous applications in epidemiology, environmental monitoring, and homeland security. SSS provides a way to rigorously test for the existence of an anomaly and provide statistical guarantees as to how "anomalous" that anomaly is. However, these methods rely on defining specific regions where the spatial information a point contributes is limited to binary 0 or 1, of either inside or outside the region, while in reality anomalies will tend to follow smooth distributions with decaying density further from an epicenter. In this work, we propose a method that addresses this shortcoming through a continuous scan statistic that generalizes SSS by allowing the point contribution to be defined by a kernel. We provide extensive experimental and theoretical results that shows our methods can be computed efficiently while providing high statistical power for detecting anomalous regions.

stat.ML↗

arXiv2019

Sketched MinDist

We consider sketch vectors of geometric objects $J$ through the \mindist function \[ v_i(J) = \inf_{p \in J} \|p-q_i\| \] for $q_i \in Q$ from a point set $Q$. Collecting the vector of these sketch values induces a simple, effective, and powerful distance: the Euclidean distance between these sketched vectors. This paper shows how large this set $Q$ needs to be under a variety of shapes and scenarios. For hyperplanes we provide direct connection to the sensitivity sample framework, so relative error can be preserved in $d$ dimensions using $Q = O(d/\varepsilon^2)$. However, for other shapes, we show we need to enforce a minimum distance parameter $ρ$, and a domain size $L$. For $d=2$ the sample size $Q$ then can be $\tilde{O}((L/ρ) \cdot 1/\varepsilon^2)$. For objects (e.g., trajectories) with at most $k$ pieces this can provide stronger \emph{for all} approximations with $\tilde{O}((L/ρ)\cdot k^3 / \varepsilon^2)$ points. Moreover, with similar size bounds and restrictions, such trajectories can be reconstructed exactly using only these sketch vectors.

cs.CG↗

arXiv2019

Simple Distances for Trajectories via Landmarks

We develop a new class of distances for objects including lines, hyperplanes, and trajectories, based on the distance to a set of landmarks. These distances easily and interpretably map objects to a Euclidean space, are simple to compute, and perform well in data analysis tasks. For trajectories, they match and in some cases significantly out-perform all state-of-the-art other metrics, can effortlessly be used in k-means clustering, and directly plugged into approximate nearest neighbor approaches which immediately out-perform the best recent advances in trajectory similarity search by several orders of magnitude. These distances do not require a geometry distorting dual (common in the line or halfspace case) or complicated alignment (common in trajectory case). We show reasonable and often simple conditions under which these distances are metrics.

cs.CG↗

arXiv2019

Scalable Spatial Scan Statistics for Trajectories

We define several new models for how to define anomalous regions among enormous sets of trajectories. These are based on spatial scan statistics, and identify a geometric region which captures a subset of trajectories which are significantly different in a measured characteristic from the background population. The model definition depends on how much a geometric region is contributed to by some overlapping trajectory. This contribution can be the full trajectory, proportional to the time spent in the spatial region, or dependent on the flux across the boundary of that spatial region. Our methods are based on and significantly extend a recent two-level sampling approach which provides high accuracy at enormous scales of data. We support these new models and algorithms with extensive experiments on millions of trajectories and also theoretical guarantees.

cs.DS↗

arXiv2019

Near-Optimal Coresets of Kernel Density Estimates

We construct near-optimal coresets for kernel density estimates for points in $\mathbb{R}^d$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(\sqrt{d}/\varepsilon\cdot \sqrt{\log 1/\varepsilon} )$, and we show a near-matching lower bound of size $Ω(\min\{\sqrt{d}/\varepsilon, 1/\varepsilon^2\})$. When $d\geq 1/\varepsilon^2$, it is known that the size of coreset can be $O(1/\varepsilon^2)$. The upper bound is a polynomial-in-$(1/\varepsilon)$ improvement when $d \in [3,1/\varepsilon^2)$ and the lower bound is the first known lower bound to depend on $d$ for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.

cs.LG↗

arXiv2019

Independent Range Sampling, Revisited Again

We revisit the range sampling problem: the input is a set of points where each point is associated with a real-valued weight. The goal is to store them in a structure such that given a query range and an integer $k$, we can extract $k$ independent random samples from the points inside the query range, where the probability of sampling a point is proportional to its weight. This line of work was initiated in 2014 by Hu, Qiao, and Tao and it was later followed up by Afshani and Wei. The first line of work mostly studied unweighted but dynamic version of the problem in one dimension whereas the second result considered the static weighted problem in one dimension as well as the unweighted problem in 3D for halfspace queries. We offer three main results and some interesting insights that were missed by the previous work: We show that it is possible to build efficient data structures for range sampling queries if we allow the query time to hold in expectation (the first result), or obtain efficient worst-case query bounds by allowing the sampling probability to be approximately proportional to the weight (the second result). The third result is a conditional lower bound that shows essentially one of the previous two concessions is needed. For instance, for the 3D range sampling queries, the first two results give efficient data structures with near-linear space and polylogarithmic query time whereas the lower bound shows with near-linear space the worst-case query time must be close to $n^{2/3}$, ignoring polylogarithmic factors. Up to our knowledge, this is the first such major gap between the expected and worst-case query time of a range searching problem.

cs.DS↗

arXiv2018

Computing Approximate Statistical Discrepancy

Consider a geometric range space $(X,cA)$ where each data point $x \in X$ has two or more values (say $r(x)$ and $b(x)$). Also consider a function $Φ(A)$ defined on any subset $A \in (X,cA)$ on the sum of values in that range e.g., $r_A = \sum_{x \in A} r(x)$ and $b_A = \sum_{x \in A} b(x)$. The $Φ$-maximum range is $A^* = \arg \max_{A \in (X,cA)} Φ(A)$. Our goal is to find some $\hat{A}$ such that $|Φ(\hat{A}) - Φ(A^*)| \leq \varepsilon.$ We develop algorithms for this problem for range spaces with bounded VC-dimension, as well as significant improvements for those defined by balls, halfspaces, and axis-aligned rectangles. This problem has many applications in many areas including discrepancy evaluation, classification, and spatial scan statistics.

cs.CG↗

arXiv2018

Improved Bounds on Information Dissemination by Manhattan Random Waypoint Model

With the popularity of portable wireless devices it is important to model and predict how information or contagions spread by natural human mobility -- for understanding the spreading of deadly infectious diseases and for improving delay tolerant communication schemes. Formally, we model this problem by considering $M$ moving agents, where each agent initially carries a \emph{distinct} bit of information. When two agents are at the same location or in close proximity to one another, they share all their information with each other. We would like to know the time it takes until all bits of information reach all agents, called the \textit{flood time}, and how it depends on the way agents move, the size and shape of the network and the number of agents moving in the network. We provide rigorous analysis for the \MRWP model (which takes paths with minimum number of turns), a convenient model used previously to analyze mobile agents, and find that with high probability the flood time is bounded by $O\big(N\log M\lceil(N/M) \log(NM)\rceil\big)$, where $M$ agents move on an $N\times N$ grid. In addition to extensive simulations, we use a data set of taxi trajectories to show that our method can successfully predict flood times in both experimental settings and the real world.

cs.MA↗

arXiv2018

Practical Low-Dimensional Halfspace Range Space Sampling

We develop, analyze, implement, and compare new algorithms for creating $\varepsilon$-samples of range spaces defined by halfspaces which have size sub-quadratic in $1/\varepsilon$, and have runtime linear in the input size and near-quadratic in $1/\varepsilon$. The key to our solution is an efficient construction of partition trees. Despite not requiring any techniques developed after the early 1990s, apparently such a result was not ever explicitly described. We demonstrate that our implementations, including new implementations of several variants of partition trees, do indeed run in time linear in the input, appear to run linear in output size, and observe smaller error for the same size sample compared to the ubiquitous random sample (which requires size quadratic in $1/\varepsilon$). This result has direct applications in speeding up discrepancy evaluation, approximate range counting, and spatial anomaly detection.

cs.CG↗

arXiv2018

Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data

Robust estimators, like the median of a point set, are important for data analysis in the presence of outliers. We study robust estimators for locationally uncertain points with discrete distributions. That is, each point in a data set has a discrete probability distribution describing its location. The probabilistic nature of uncertain data makes it challenging to compute such estimators, since the true value of the estimator is now described by a distribution rather than a single point. We show how to construct and estimate the distribution of the median of a point set. Building the approximate support of the distribution takes near-linear time, and assigning probability to that support takes quadratic time. We also develop a general approximation technique for distributions of robust estimators with respect to ranges with bounded VC dimension. This includes the geometric median for high dimensions and the Siegel estimator for linear regression.

cs.DM↗

arXiv2017

Visualizing Sensor Network Coverage with Location Uncertainty

We present an interactive visualization system for exploring the coverage in sensor networks with uncertain sensor locations. We consider a simple case of uncertainty where the location of each sensor is confined to a discrete number of points sampled uniformly at random from a region with a fixed radius. Employing techniques from topological data analysis, we model and visualize network coverage by quantifying the uncertainty defined on its simplicial complex representations. We demonstrate the capabilities and effectiveness of our tool via the exploration of randomly distributed sensor networks.

cs.HC↗

arXiv2017

Improved Coresets for Kernel Density Estimates

We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic kernels (including Gaussian and Laplace kernels), our approximation preserves the $L_\infty$ error between kernel density estimates within error $ε$, with coreset size $2/ε^2$, but no other aspects of the data, including the dimension, the diameter of the point set, or the bandwidth of the kernel common to other approximations. When the dimension is unrestricted, we show this bound is tight for these kernels as well as a much broader set. This work provides a careful analysis of the iterative Frank-Wolfe algorithm adapted to this context, an algorithm called \emph{kernel herding}. This analysis unites a broad line of work that spans statistics, machine learning, and geometry. When the dimension $d$ is constant, we demonstrate much tighter bounds on the size of the coreset specifically for Gaussian kernels, showing that it is bounded by the size of the coreset for axis-aligned rectangles. Currently the best known constructive bound is $O(\frac{1}ε \log^d \frac{1}ε)$, and non-constructively, this can be improved by $\sqrt{\log \frac{1}ε}$. This improves the best constant dimension bounds polynomially for $d \geq 3$.

cs.LG↗

arXiv2017

Visualization of Big Spatial Data using Coresets for Kernel Density Estimates

The size of large, geo-located datasets has reached scales where visualization of all data points is inefficient. Random sampling is a method to reduce the size of a dataset, yet it can introduce unwanted errors. We describe a method for subsampling of spatial data suitable for creating kernel density estimates from very large data and demonstrate that it results in less error than random sampling. We also introduce a method to ensure that thresholding of low values based on sampled data does not omit any regions above the desired threshold when working with sampled data. We demonstrate the effectiveness of our approach using both, artificial and real-world large geospatial datasets.

cs.HC↗