21,586 research outputs found
Geometry Helps to Compare Persistence Diagrams
Exploiting geometric structure to improve the asymptotic complexity of
discrete assignment problems is a well-studied subject. In contrast, the
practical advantages of using geometry for such problems have not been
explored. We implement geometric variants of the Hopcroft--Karp algorithm for
bottleneck matching (based on previous work by Efrat el al.) and of the auction
algorithm by Bertsekas for Wasserstein distance computation. Both
implementations use k-d trees to replace a linear scan with a geometric
proximity query. Our interest in this problem stems from the desire to compute
distances between persistence diagrams, a problem that comes up frequently in
topological data analysis. We show that our geometric matching algorithms lead
to a substantial performance gain, both in running time and in memory
consumption, over their purely combinatorial counterparts. Moreover, our
implementation significantly outperforms the only other implementation
available for comparing persistence diagrams.Comment: 20 pages, 10 figures; extended version of paper published in ALENEX
201
High-dimensional approximate nearest neighbor: k-d Generalized Randomized Forests
We propose a new data-structure, the generalized randomized kd forest, or
kgeraf, for approximate nearest neighbor searching in high dimensions. In
particular, we introduce new randomization techniques to specify a set of
independently constructed trees where search is performed simultaneously, hence
increasing accuracy. We omit backtracking, and we optimize distance
computations, thus accelerating queries. We release public domain software
geraf and we compare it to existing implementations of state-of-the-art methods
including BBD-trees, Locality Sensitive Hashing, randomized kd forests, and
product quantization. Experimental results indicate that our method would be
the method of choice in dimensions around 1,000, and probably up to 10,000, and
pointsets of cardinality up to a few hundred thousands or even one million;
this range of inputs is encountered in many critical applications today. For
instance, we handle a real dataset of images represented in 960
dimensions with a query time of less than sec on average and 90\% responses
being true nearest neighbors
DROP: Dimensionality Reduction Optimization for Time Series
Dimensionality reduction is a critical step in scaling machine learning
pipelines. Principal component analysis (PCA) is a standard tool for
dimensionality reduction, but performing PCA over a full dataset can be
prohibitively expensive. As a result, theoretical work has studied the
effectiveness of iterative, stochastic PCA methods that operate over data
samples. However, termination conditions for stochastic PCA either execute for
a predetermined number of iterations, or until convergence of the solution,
frequently sampling too many or too few datapoints for end-to-end runtime
improvements. We show how accounting for downstream analytics operations during
DR via PCA allows stochastic methods to efficiently terminate after operating
over small (e.g., 1%) subsamples of input data, reducing whole workload
runtime. Leveraging this, we propose DROP, a DR optimizer that enables speedups
of up to 5x over Singular-Value-Decomposition-based PCA techniques, and exceeds
conventional approaches like FFT and PAA by up to 16x in end-to-end workloads
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