Publications

Results 26–50 of 75
Skip to search filters

Geometric comparison of popular mixture-model distances

Mitchell, Scott A.

Statistical Latent Dirichlet Analysis produces mixture model data that are geometrically equivalent to points lying on a regular simplex in moderate to high dimensions. Numerous other statistical models and techniques also produce data in this geometric category, even though the meaning of the axes and coordinate values differs significantly. A distance function is used to further analyze these points, for example to cluster them. Several different distance functions are popular amongst statisticians; which distance function is chosen is usually driven by the historical preference of the application domain, information-theoretic considerations, or by the desirability of the clustering results. Relatively little consideration is usually given to how distance functions geometrically transform data, or the distances algebraic properties. Here we take a look at these issues, in the hope of providing complementary insight and inspiring further geometric thought. Several popular distances, {chi}{sup 2}, Jensen - Shannon divergence, and the square of the Hellinger distance, are shown to be nearly equivalent; in terms of functional forms after transformations, factorizations, and series expansions; and in terms of the shape and proximity of constant-value contours. This is somewhat surprising given that their original functional forms look quite different. Cosine similarity is the square of the Euclidean distance, and a similar geometric relationship is shown with Hellinger and another cosine. We suggest a geodesic variation of Hellinger. The square-root projection that arises in Hellinger distance is briefly compared to standard normalization for Euclidean distance. We include detailed derivations of some ratio and difference bounds for illustrative purposes. We provide some constructions that nearly achieve the worst-case ratios, relevant for contours.

More Details
TYPE Conference YEAR 2010
OSTI

LDRD 149045 final report distinguishing documents

Mitchell, Scott A.

This LDRD 149045 final report describes work that Sandians Scott A. Mitchell, Randall Laviolette, Shawn Martin, Warren Davis, Cindy Philips and Danny Dunlavy performed in 2010. Prof. Afra Zomorodian provided insight. This was a small late-start LDRD. Several other ongoing efforts were leveraged, including the Networks Grand Challenge LDRD, and the Computational Topology CSRF project, and the some of the leveraged work is described here. We proposed a sentence mining technique that exploited both the distribution and the order of parts-of-speech (POS) in sentences in English language documents. The ultimate goal was to be able to discover 'call-to-action' framing documents hidden within a corpus of mostly expository documents, even if the documents were all on the same topic and used the same vocabulary. Using POS was novel. We also took a novel approach to analyzing POS. We used the hypothesis that English follows a dynamical system and the POS are trajectories from one state to another. We analyzed the sequences of POS using support vector machines and the cycles of POS using computational homology. We discovered that the POS were a very weak signal and did not support our hypothesis well. Our original goal appeared to be unobtainable with our original approach. We turned our attention to study an aspect of a more traditional approach to distinguishing documents. Latent Dirichlet Allocation (LDA) turns documents into bags-of-words then into mixture-model points. A distance function is used to cluster groups of points to discover relatedness between documents. We performed a geometric and algebraic analysis of the most popular distance functions and made some significant and surprising discoveries, described in a separate technical report.

More Details
TYPE SAND Report YEAR 2010
OSTIDOI

Meshes optimized for discrete exterior calculus (DEC)

Mitchell, Scott A.; Mousley, Sarah C.; Deakin, Michael D.; Knupp, Patrick M.; Mitchell, Scott A.

We study the optimization of an energy function used by the meshing community to measure and improve mesh quality. This energy is non-traditional because it is dependent on both the primal triangulation and its dual Voronoi (power) diagram. The energy is a measure of the mesh's quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. The energy of a mesh gives an upper bound on the error of the discrete diagonal approximation of the Hodge star operator. In practice, one begins with an initial mesh and then makes adjustments to produce a mesh of lower energy. However, we have discovered several shortcomings in directly optimizing this energy, e.g. its non-convexity, and we show that the search for an optimized mesh may lead to mesh inversion (malformed triangles). We propose a new energy function to address some of these issues.

More Details
TYPE Other Report YEAR 2017
OSTIDOI

Methods for Multisweep Automation

Shepherd, Jason F.; Mitchell, Scott A.; Knupp, Patrick K.; Mitchell, Scott A.

Sweeping has become the workhorse algorithm for creating conforming hexahedral meshes of complex models. This paper describes progress on the automatic, robust generation of MultiSwept meshes in CUBIT. MultiSweeping extends the class of volumes that may be swept to include those with multiple source and multiple target surfaces. While not yet perfect, CUBIT's MultiSweeping has recently become more reliable, and been extended to assemblies of volumes. Sweep Forging automates the process of making a volume (multi) sweepable: Sweep Verification takes the given source and target surfaces, and automatically classifies curve and vertex types so that sweep layers are well formed and progress from sources to targets.

More Details
TYPE Conference YEAR 2000
OSTI

POF-Darts: Geometric adaptive sampling for probability of failure

Reliability Engineering and System Safety

Ebeida, Mohamed S.; Mitchell, Scott A.; Swiler, Laura P.; Romero, Vicente J.; Rushdi, Ahmad A.

We introduce a novel technique, POF-Darts, to estimate the Probability Of Failure based on random disk-packing in the uncertain parameter space. POF-Darts uses hyperplane sampling to explore the unexplored part of the uncertain space. We use the function evaluation at a sample point to determine whether it belongs to failure or non-failure regions, and surround it with a protection sphere region to avoid clustering. We decompose the domain into Voronoi cells around the function evaluations as seeds and choose the radius of the protection sphere depending on the local Lipschitz continuity. As sampling proceeds, regions uncovered with spheres will shrink, improving the estimation accuracy. After exhausting the function evaluation budget, we build a surrogate model using the function evaluations associated with the sample points and estimate the probability of failure by exhaustive sampling of that surrogate. In comparison to other similar methods, our algorithm has the advantages of decoupling the sampling step from the surrogate construction one, the ability to reach target POF values with fewer samples, and the capability of estimating the number and locations of disconnected failure regions, not just the POF value. We present various examples to demonstrate the efficiency of our novel approach.

More Details
TYPE Journal Article YEAR 2016
ScopusOSTIDOI
Results 26–50 of 75
Results 26–50 of 75