12,962 research outputs found
Exact dimensionality and projections of random self-similar measures and sets
We study the geometric properties of random multiplicative cascade measures
defined on self-similar sets. We show that such measures and their projections
and sections are almost surely exact-dimensional, generalizing Feng and Hu's
result \cite{FeHu09} for self-similar measures. This, together with a compact
group extension argument, enables us to generalize Hochman and Shmerkin's
theorems on projections of deterministic self-similar measures \cite{HoSh12} to
these random measures without requiring any separation conditions on the
underlying sets. We give applications to self-similar sets and fractal
percolation, including new results on projections, -images and distance
sets.Comment: 25 page
Indexability, concentration, and VC theory
Degrading performance of indexing schemes for exact similarity search in high
dimensions has long since been linked to histograms of distributions of
distances and other 1-Lipschitz functions getting concentrated. We discuss this
observation in the framework of the phenomenon of concentration of measure on
the structures of high dimension and the Vapnik-Chervonenkis theory of
statistical learning.Comment: 17 pages, final submission to J. Discrete Algorithms (an expanded,
improved and corrected version of the SISAP'2010 invited paper, this e-print,
v3
FLASH: Randomized Algorithms Accelerated over CPU-GPU for Ultra-High Dimensional Similarity Search
We present FLASH (\textbf{F}ast \textbf{L}SH \textbf{A}lgorithm for
\textbf{S}imilarity search accelerated with \textbf{H}PC), a similarity search
system for ultra-high dimensional datasets on a single machine, that does not
require similarity computations and is tailored for high-performance computing
platforms. By leveraging a LSH style randomized indexing procedure and
combining it with several principled techniques, such as reservoir sampling,
recent advances in one-pass minwise hashing, and count based estimations, we
reduce the computational and parallelization costs of similarity search, while
retaining sound theoretical guarantees.
We evaluate FLASH on several real, high-dimensional datasets from different
domains, including text, malicious URL, click-through prediction, social
networks, etc. Our experiments shed new light on the difficulties associated
with datasets having several million dimensions. Current state-of-the-art
implementations either fail on the presented scale or are orders of magnitude
slower than FLASH. FLASH is capable of computing an approximate k-NN graph,
from scratch, over the full webspam dataset (1.3 billion nonzeros) in less than
10 seconds. Computing a full k-NN graph in less than 10 seconds on the webspam
dataset, using brute-force (), will require at least 20 teraflops. We
provide CPU and GPU implementations of FLASH for replicability of our results
- …