Approximation and Streaming Algorithms for Projective Clustering via Random Projections

Let $P$ be a set of $n$ points in $\mathbb{R}^d$. In the projective clustering problem, given $k, q$ and norm $\rho \in [1,\infty]$, we have to compute a set $\mathcal{F}$ of $k$ $q$-dimensional flats such that $(\sum_{p\in P}d(p, \mathcal{F})^\rho)^{1/\rho}$ is minimized; here $d(p, \mathcal{F})$ represents the (Euclidean) distance of $p$ to the closest flat in $\mathcal{F}$. We let $f_k^q(P,\rho)$ denote the minimal value and interpret $f_k^q(P,\infty)$ to be $\max_{r\in P}d(r, \mathcal{F})$. When $\rho=1,2$ and $\infty$ and $q=0$, the problem corresponds to the $k$-median, $k$-mean and the $k$-center clustering problems respectively. For every $0 < \epsilon < 1$, $S\subset P$ and $\rho \ge 1$, we show that the orthogonal projection of $P$ onto a randomly chosen flat of dimension $O(((q+1)^2\log(1/\epsilon)/\epsilon^3) \log n)$ will $\epsilon$-approximate $f_1^q(S,\rho)$. This result combines the concepts of geometric coresets and subspace embeddings based on the Johnson-Lindenstrauss Lemma. As a consequence, an orthogonal projection of $P$ to an $O(((q+1)^2 \log ((q+1)/\epsilon)/\epsilon^3) \log n)$ dimensional randomly chosen subspace $\epsilon$-approximates projective clusterings for every $k$ and $\rho$ simultaneously. Note that the dimension of this subspace is independent of the number of clusters~$k$. Using this dimension reduction result, we obtain new approximation and streaming algorithms for projective clustering problems. For example, given a stream of $n$ points, we show how to compute an $\epsilon$-approximate projective clustering for every $k$ and $\rho$ simultaneously using only $O((n+d)((q+1)^2\log ((q+1)/\epsilon))/\epsilon^3 \log n)$ space. Compared to standard streaming algorithms with $\Omega(kd)$ space requirement, our approach is a significant improvement when the number of input points and their dimensions are of the same order of magnitude.Comment: Canadian Conference on Computational Geometry (CCCG 2015

In-lab three-dimensional printing:an inexpensive tool for experimentation and visualization for the field of organogenesis

The development of the microscope in 1590 by Zacharias Janssenby and Hans Lippershey gave the world a new way of visualizing details of morphogenesis and development. More recent improvements in this technology including confocal microscopy, scanning electron microscopy (SEM) and optical projection tomography (OPT) have enhanced the quality of the resultant image. These technologies also allow a representation to be made of a developing tissue’s three-dimensional (3-D) form. With all these techniques however, the image is delivered on a flat two-dimensional (2-D) screen. 3-D printing represents an exciting potential to reproduce the image not simply on a flat screen, but in a physical, palpable three-dimensional structure. Here we explore the scope that this holds for exploring and interacting with the structure of a developing organ in an entirely novel way. As well as being useful for visualization, 3-D printers are capable of rapidly and cost-effectively producing custom-made structures for use within the laboratory. We here describe the advantages of producing hardware for a tissue culture system using an inexpensive in-lab printer

COLOR MULTIPLEXED SINGLE PATTERN SLI

Structured light pattern projection techniques are well known methods of accurately capturing 3-Dimensional information of the target surface. Traditional structured light methods require several different patterns to recover the depth, without ambiguity or albedo sensitivity, and are corrupted by object movement during the projection/capture process. This thesis work presents and discusses a color multiplexed structured light technique for recovering object shape from a single image thus being insensitive to object motion. This method uses single pattern whose RGB channels are each encoded with a unique subpattern. The pattern is projected on to the target and the reflected image is captured using high resolution color digital camera. The image is then separated into individual color channels and analyzed for 3-D depth reconstruction through use of phase decoding and unwrapping algorithms thereby establishing the viability of the color multiplexed single pattern technique. Compared to traditional methods (like PMP, Laser Scan etc) only one image/one-shot measurement is required to obtain the 3-D depth information of the object, requires less expensive hardware and normalizes albedo sensitivity and surface color reflectance variations. A cosine manifold and a flat surface are measured with sufficient accuracy demonstrating the feasibility of a real-time system

Binary black hole spacetimes with a helical Killing vector

Binary black hole spacetimes with a helical Killing vector, which are discussed as an approximation for the early stage of a binary system, are studied in a projection formalism. In this setting the four dimensional Einstein equations are equivalent to a three dimensional gravitational theory with a $SL(2,\mathbb{C})/SO(1,1)$ sigma model as the material source. The sigma model is determined by a complex Ernst equation. 2+1 decompositions of the 3-metric are used to establish the field equations on the orbit space of the Killing vector. The two Killing horizons of spherical topology which characterize the black holes, the cylinder of light where the Killing vector changes from timelike to spacelike, and infinity are singular points of the equations. The horizon and the light cylinder are shown to be regular singularities, i.e. the metric functions can be expanded in a formal power series in the vicinity. The behavior of the metric at spatial infinity is studied in terms of formal series solutions to the linearized Einstein equations. It is shown that the spacetime is not asymptotically flat in the strong sense to have a smooth null infinity under the assumption that the metric tends asymptotically to the Minkowski metric. In this case the metric functions have an oscillatory behavior in the radial coordinate in a non-axisymmetric setting, the asymptotic multipoles are not defined. The asymptotic behavior of the Weyl tensor near infinity shows that there is no smooth null infinity.Comment: to be published in Phys. Rev. D, minor correction

Approximation and Streaming Algorithms for Projective Clustering via Random Projections

Abstract Let P be a set of n points in R d . In the projective clustering problem, given k, q and norm ρ ∈ [1, ∞], we have to compute a set F of k q-dimensional flats such that represents the (Euclidean) distance of p to the closest flat in F. We let f q k (P, ρ) denote the minimal value and interpret f q k (P, ∞) to be max r∈P d(r, F). When ρ = 1, 2 and ∞ and q = 0, the problem corresponds to the k-median, kmean and the k-center clustering problems respectively. For every 0 &lt; ε &lt; 1, S ⊂ P and ρ ≥ 1, we show that the orthogonal projection of P onto a randomly chosen flat of dimension O(((q + 1) 2 log(1/ε)/ε 3 ) log n) will ε-approximate f q 1 (S, ρ). This result combines the concepts of geometric coresets and subspace embeddings based on the Johnson-Lindenstrauss Lemma. As a consequence, an orthogonal projection of P to an O(((q + 1) 2 log((q + 1)/ε)/ε 3 ) log n) dimensional randomly chosen subspace ε-approximates projective clusterings for every k and ρ simultaneously. Note that the dimension of this subspace is independent of the number of clusters k. Using this dimension reduction result, we obtain new approximation and streaming algorithms for projective clustering problems. For example, given a stream of n points, we show how to compute an ε-approximate projective clustering for every k and ρ simultaneously using only O((n + d)((q + 1) 2 log((q + 1)/ε))/ε 3 log n) space. Compared to standard streaming algorithms with Ω(kd) space requirement, our approach is a significant improvement when the number of input points and their dimensions are of the same order of magnitude