Seattle Scalability Mahout

Numerical Recipes
in
Hadoop
Jake Mannix
linkedin/in/jakemannix
twitter/pbrane
jake.mannix@gmail.com
jmannix@apache.org
Principal SDE, LinkedIn
Committer, Apache Mahout, Zoie, Bobo-Browse, Decomposer
Author, Lucene in Depth (Manning MM/DD/2010)

A Mathematician’s Apology
What mathematical structure describes all of these?
Full-text search:
Score documents matching “query string”
Collaborative filtering recommendation:
Users who liked {those} also liked {these}
(Social/web)-graph proximity:
People/pages “close” to {this} are {these}

Matrix Multiplication!

Full-text Search
Vector Space Model of IR
Corpus as term-document matrix
Query as bag-of-words vector
Full-text search is just:

Collaborative Filtering
User preference matrix
(and item-item similarity matrix )
Input user as vector of preferences
(simple) Item-based CF recommendations are:
T

Graph Proximity
Adjacency matrix:
2nd degree adjacency matrix:
Input all of a user’s “friends” or page links:
(weighted) distance measure of 1st – 3rd degree connections is then:

Dictionary
Applications Linear Algebra

How does this help?
In Search:
Latent Semantic Indexing (LSI)
probabalistic LSI
Latent Dirichlet Allocation
In Recommenders:
Singular Value Decomposition
Layered Restricted Boltzmann Machines
(Deep Belief Networks)
In Graphs:
PageRank
Spectral Decomposition / Spectral Clustering

Often use “Dimensional Reduction”
To alleviate the sparse Big Data problem of “the curse of dimensionality”
Used to improve recall and relevance
in general: smooth the metric on your data set

New applications with Matrices
If Search is finding doc-vector by:
and users query with data represented: Q =
Giving implicit feedback based on click-through per session: C =

… continued
Then has the form (docs-by-terms) for search!
Approach has been used by Ted Dunning at Veoh
(and probably others)

Linear Algebra performance tricks
Naïve item-based recommendations:
Calculate item similarity matrix:
Calculate item recs:
Express in one step:
In matrix notation:
Re-writing as:
is the vector of preferences for user “v”,
is the vector of preferences of item “i”
The result is the matrix sum of the outer (tensor) products of these vectors, scaled by the entry they intersect at.

Item Recommender via Hadoop

Apache Mahout
Apache Mahout currently on release 0.3
http://lucene.apache.org/mahout
Will be a “Top Level Project” soon (before 0.4)
( http://mahout.apache.org )
“Scalable Machine Learning with commercially friendly licensing”

Mahout Features
Recommenders
absorbed the Taste project
Classification (Naïve Bayes, C-Bayes, more)
Clustering (Canopy, fuzzy-K-means, Dirichlet, etc…)
Fast non-distributed linear mathematics
absorbed the classic CERN Colt project
Distributed Matrices and decomposition
absorbed the Decomposer project
mahout shell-script analogous to $HADOOP_HOME/bin/hadoop
$MAHOUT_HOME/bin/mahout kmeans –i “in” –o “out” –k 100
$MAHOUT_HOME/bin/mahout svd –i “in” –o “out” –k 300
etc…
Taste web-app for real-time recommendations

DistributedRowMatrix
Wrapper around a SequenceFile<IntWritable,VectorWritable>
Distributed methods like:
Matrix transpose();
Matrix times(Matrix other);
Vector times(Vectorv);
Vector timesSquared(Vectorv);
To get SVD: pass into DistributedLanczosSolver:
LanczosSolver.solve(Matrix input, Matrix eigenVectors, List<Double> eigenValues, int rank);

Questions?
Contact:
jake.mannix@gmail.com
jmannix@apache.org
http://twitter.com/pbrane
http://www.decomposer.org/blog
http://www.linkedin.com/in/jakemannix

Appendix
There are lots of ways to deal with sparse Big Data, and many (not all) need to deal with the dimensionality of the feature-space growing beyond reasonable limits, and techniques to deal with this depend heavily on your data…
That having been said, there are some general techniques

Dealing with Curse of Dimensionality
Sparseness means fast, but overlap is too small
Can we reduce the dimensionality (from “all possible text tokens” or “all userIds”) while keeping the nice aspects of the search problem?
If possible, collapse “similar” vectors (synonymous terms, userIds with high overlap, etc…) towards each other while keeping “dissimilar” vectors far apart…

Solution A: Matrix decomposition
Singular Value Decomposition (truncated)
“best” approximation to your matrix
Used in Latent Semantic Indexing (LSI)
For graphs: spectral decomposition
Collaborative filtering (Netflix leaderboard)
Issues: very computation intensive
no parallelized open-source packages see Apache Mahout
Makes things too dense

SVD: continued
Hadoopimpl. in Mahout (Lanczos)
O(N*d*k) for rank-k SVD on N docs, delt’s each
Density can be dealt with by doing Canopy Clustering offline
But only extracting linear feature mixes
Also, still very computation intensive and I/O intensive (k-passes over data set), are there better dimensional reduction methods?

Solution B: Stochastic Decomposition co-ocurrence-based kernel + online Random Projection + SVD

Co-ocurrence-based kernel
Extract bigram phrases / pairs of items rated by the same person (using Log-Likelihood Ratio test to pick the best)
“Disney on Ice was Amazing!” -> {“disney”, “disney on ice”, “ice”, “was” “amazing”}
{item1:4, item2:5, item5:3, item9:1} -> {item1:4, (items1+2):4.5, item2:5, item5:3,…}
Dim(features) goes from 105to 108+(yikes!)

Online Random Projection
Randomly project kernelized text vectors down to “merely” 103dimensions with a Gaussian matrix
Or project eachnGram down to an random (but sparse) 103-dim vector:
V= {123876244 =>1.3} (tf-IDF of “disney”)
V’= c*{h(i) => 1, h(h(i)) =>1, h(h(h(i))) =>1}
(c= 1.3 / sqrt(3))

Outer-product and Sum
Take the 103-dim projected vectors and outer-product with themselves,
result is 103x103-dim matrix
sum these in a Combiner
All results go to single Reducer, where you compute…

SVD
SVD-them quickly (they fit in memory)
Over and over again (as new data comes in)
Use the most recent SVD to project your (already randomly projected) text still further (now encoding “semantic” similarity).
SVD-projected vectors can be assigned immediately to nearest clusters if desired

References
Randomized matrix decomposition review: http://arxiv.org/abs/0909.4061
Sparse hashing/projection:
John Langford et al. “VowpalWabbit”
http://hunch.net/~vw/

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Seattle Scalability Mahout

by Jake Mannix

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