Jeff M. Phillips

Outlier Robust ICP for Minimizing Fractional RMSD

We describe a variation of the iterative closest point (ICP) algorithm for aligning two point sets under a set of transformations. Our algorithm is superior to previous algorithms because (1) in determining the optimal alignment, it identifies and discards likely outliers in a statistically robust manner, and (2) it is guaranteed to converge to a locally optimal solution. To this end, we formalize a new distance measure, fractional root mean squared distance (FRMSD), which incorporates the fraction of inliers into the distance function. Our framework can easily incorporate most techniques and heuristics from modern registration algorithms. We experimentally validate our algorithm against previous techniques on 2 and 3 dimensional data exposed to a variety of outlier types.

Simulated knot tying

Applications such as suturing in medical simulations require the modeling of knot tying in physically realistic rope. The paper describes the design and implementation of such a system. Our model uses a spline of linear springs, adaptive subdivision and a dynamics simulation. Collisions are discrete event simulated and follow the impulse model. Although some care must be taken to maintain stable knots, we demonstrate our simple model is sufficient for this task. In particular, we do not use friction or explicit constraints to maintain the knot. As examples, we tie an overhand knot and a reef knot.

Radio tomographic imaging and tracking of stationary and moving people via kernel distance

Network radio frequency (RF) environment sensing (NRES) systems pinpoint and track people in buildings using changes in the signal strength measurements made by a wireless sensor network. It has been shown that such systems can locate people who do not participate in the system by wearing any radio device, even through walls, because of the changes that moving people cause to the static wireless sensor network. However, many such systems cannot locate stationary people. We present and evaluate a system which can locate stationary or moving people, without calibration, by using kernel distance to quantify the difference between two histograms of signal strength measurements. From five experiments, we show that our kernel distance-based radio tomographic localization system performs better than the state-of-the-art NRES systems in different non line-of-sight environments.

Nearest neighbor searching under uncertainty II

Nearest-neighbor queries, which ask for returning the nearest neighbor of a query point in a set of points, are important and widely studied in many fields because of a wide range of applications. In many of these applications, such as sensor databases, location based services, face recognition, and mobile data, the location of data is imprecise. We therefore study nearest neighbor queries in a probabilistic framework in which the location of each input point and/or query point is specified as a probability density function and the goal is to return the point that minimizes the expected distance, which we refer to as the expected nearest neighbor (ENN). We present methods for computing an exact ENN or an ε-approximate ENN, for a given error parameter 0 < ε 0 < 1, under different distance functions. These methods build an index of near-linear size and answer ENN queries in polylogarithmic or sublinear time, depending on the underlying function. As far as we know, these are the first nontrivial methods for answering exact or ε-approximate ENN queries with provable performance guarantees.

Guided Expansive Spaces Trees: a search strategy for motion- and cost-constrained state spaces

Motion planning for systems with constraints on controls or the need for relatively straight paths for real-time actions presents challenges for modern planners. This paper presents an approach which addresses these types of systems by building on existing motion planning approaches. Guided Expansive Spaces Trees are introduced to search for a low cost and relatively straight path in a space with motion constraints. Path Gradient Descent, which builds on the idea of Elastic Strips, finds the locally optimal path for an existing path. These techniques are tested on simulations of rendezvous and docking of the space shuttle to the International Space Station and of a 4-foot fan-controlled blimp in a factory setting.

The hunting of the bump: on maximizing statistical discrepancy

Anomaly detection has important applications in biosurveilance and environmental monitoring. When comparing measured data to data drawn from a baseline distribution, merely, finding clusters in the measured data may not actually represent true anomalies. These clusters may likely be the clusters of the baseline distribution. Hence, a discrepancy function is often used to examine how different measured data is to baseline data within a region. An anomalous region is thus defined to be one with high discrepancy.In this paper, we present algorithms for maximizing statistical discrepancy functions over the space of axis-parallel rectangles. We give provable approximation guarantees, both additive and relative, and our methods apply to any convex discrepancy function. Our algorithms work by connecting statistical discrepancy to combinatorial discrepancy; roughly speaking, we show that in order to maximize a convex discrepancy function over a class of shapes, one needs only maximize a linear discrepancy function over the same set of shapes.We derive general discrepancy functions for data generated from a one- parameter exponential family. This generalizes the widely-used Kulldorff scan statistic for data from a Poisson distribution. We present an algorithm running in O(1/ε n2 log2n) that computes the maximum discrepancy rectangle to within additive error ε, for the Kulldorff scan statistic. Similar results hold for relative error and for discrepancy functions for data coming from Gaussian, Bernoulli, and gamma distributions. Prior to our work, the best known algorithms were exact and ran in time O(n4).

Frequent Directions: Simple and Deterministic Matrix Sketching

We describe a new algorithm called FrequentDirections for deterministic matrix sketching in the row-update model. The algorithm is presented an arbitrary input matrix A \in \mathbb{R}^{n \times d} one row at a time. It performs O(d\ell) operations per row and maintains a sketch matrix B \in \mathbb{R}^{\ell \times d} such that for any k < \ell, \|A^TA - B^TB \|_2 \leq \|A - A_k\|_F^2 / (\ell-k) {\;and\;} \|A - \pi_{B_k}(A)\|_F^2 \leq (1 + \frac{k}{\ell-k}) \|A-A_k\|_F^2. Here, A_k stands for the minimizer of \|A - A_k\|_F over all rank k matrices (similarly for B_k) and \pi_{B_k}(A) is the rank k matrix resulting from projecting A on the row span of B_k. We show that both of these bounds are the best possible for the space allowed. The summary is mergeable and hence trivially parallelizable. Moreover, FrequentDirections outperforms exemplar implementations of existing streaming algorithms in the space-error tradeoff. This paper combines, simplifies, and extends the results of Liberty [Proceedings of the 1...

Shape Fitting on Point Sets with Probability Distributions

We consider problems on data sets where each data point has uncertainty described by an individual probability distribution. We develop several frameworks and algorithms for calculating statistics on these uncertain data sets. Our examples focus on geometric shape fitting problems. We prove approximation guarantees for the algorithms with respect to the full probability distributions. We then empirically demonstrate that our algorithms are simple and practical, solving for a constant hidden by asymptotic analysis so that a user can reliably trade speed and size for accuracy.

Lower Bounds for Number-in-Hand Multiparty Communication Complexity, Made Easy

In this paper we prove lower bounds on randomized multiparty communication complexity, both in the blackboard model (where each message is written on a blackboard for all players to see) and (mainly) in the message-passing model, where messages are sent player-to-player. We introduce a new technique for proving such bounds, called symmetrization, which is natural, intuitive, and often easy to use. For example, for the problem where each of k players gets a bit-vector of length n, and the goal is to compute the coordinate-wise XOR of these vectors, we prove a tight lower bounds of Ω(nk) in the blackboard model. For the same problem with AND instead of XOR, we prove a lower bounds of roughly Ω(nk) in the message-passing model (assuming k ≤ n/3200) and Ω(n log k) in the blackboard model. We also prove lower bounds for bit-wise majority, for a graphconnectivity problem, and for other problems; the technique seems applicable to a wide range of other problems as well. The obtained communication lower bounds imply new lower bounds in the functional monitoring model [11] (also called the distributed streaming model). All of our lower bounds allow randomized communication protocols with two-sided error. We also use the symmetrization technique to prove several direct-sum-like results for multiparty communication.

Comparing distributions and shapes using the kernel distance

Starting with a similarity function between objects, it is possible to define a distance metric (the kernel distance) on pairs of objects, and more generally on probability distributions over them. These distance metrics have a deep basis in functional analysis and geometric measure theory, and have a rich structure that includes an isometric embedding into a Hilbert space. They have recently been applied to numerous problems in machine learning and shape analysis. SIn this paper, we provide the first algorithmic analysis of these distance metrics. Our main contributions are as follows: We present fast approximation algorithms for computing the kernel distance between two point sets P and Q that runs in near-linear time in the size of P ∪ Q (an explicit calculation would take quadratic time). We present polynomial-time algorithms for approximately minimizing the kernel distance under rigid transformation; they run in time O(n + poly(1/ε, log n)). We provide several general techniques for reducing complex objects to convenient sparse representations (specifically to point sets or sets of points sets) which approximately preserve the kernel distance. In particular, this allows us to reduce problems of computing the kernel distance between various types of objects such as curves, surfaces, and distributions to computing the kernel distance between point sets.