British Museum Library, LondonPicture Courtesy: flickr

Courtesy: Wikipedia

Topic Models and the Role of Sampling

Barnan Das

British Museum Library, LondonPicture Courtesy: flickr

Topic Modeling

• Methods for automatically organizing, understanding, searching and summarizing large electronic archives.• Uncover hidden topical patterns in collections.

• Annotate documents according to topics.• Using annotations to organize, summarize and search.

Topic Modeling

NIH Grants Topic Map 2011NIH Map Viewer (https://app.nihmaps.org)

Topic Modeling Applications

• Information retrieval.

• Content-based image retrieval.

• Bioinformatics

Overview of this Presentation

• Latent Dirichlet allocation (LDA)

• Approximate posterior inference• Gibbs sampling

• Paper• Fast collapsed Gibbs sampling for LDA

Latent Dirichlet Allocation

David Blei’s TalkMachine Learning Summer School, Cambridge 2009

D. M. Blei, A. Y. Ng, and M. I. Jordan, "Latent dirichlet allocation," The Journal of Machine Learning Research, vol. 3, pp. 993-1022, 2003.

Probabilistic Model

• Generative probabilistic modeling• Treats data as observations• Contains hidden variables• Hidden variables reflect thematic structure of the collection.

• Infer hidden structure using posterior inference• Discovering topics in the collection.

• Placing new data into the estimated model• Situating new documents into the estimated topic structure.

Intuition

Generative Model

Posterior Distribution

• Only documents are observable.

• Infer underlying topic structure.• Topics that generated the documents. • For each document, distribution of topics.• For each word, which topic generated the word.

• Algorithmic challenge: Finding the conditional distribution of all the latent variables, given the observation.

LDA as Graphical Model

DirichletDirichlet Multinomial Multinomial

Posterior Distribution

• From a collection of documents W, infer• Per-word topic assignment zd,n

• Per-document topic proportions d

• Per-corpus topic distribution k

• Use posterior expectation to perform different tasks.

Posterior Distribution

• Evaluate P(z|W): posterior distribution over the assignment of words to topic.

• and can be estimated.

Computing P(z|W)

• Involves evaluating a probability distribution over a large discrete space.

• Contribution of each zd,n depends on:• All z-n values.• Nk

Wn -># of times word Wd,n has been

assigned a topic k.• Nk

d -># of times a word from document d has

been assigned a topic k.

• Sampling from the target distribution using MCMC.

Approximate posterior inference:Gibbs Sampling

C. M. Bishop and SpringerLink, Pattern recognition and machine learning vol. 4: Springer New York, 2006.

Iain Murray’s TalkMachine Learning Summer School, Cambridge 2009

Overview

• When exact inference is intractable.

• Standard sampling techniques have limitation:• Cannot handle all kinds of distributions.• Cannot handle high dimensional data.

• MCMC techniques do not have these limitations.

• Markov chain:

For random variables x(1),…,x(M),

p(x(m+1)|x(1),…,x(m))=p(x(m+1)|x(m)) ; m{1,…M-1}

Gibbs Sampling

• Target distribution: p(x) = p(x1,…,xM).

• Choose the initial state of the Markov chain: {xi:i=1,…M}.

• Replace xi by a value drawn from the distribution p(xi|x-i).• xi: ith component of Z• x-i: x1,…,xM but xi omitted.

• This process is repeated for all the variables.

• Repeat the whole cycle for however many samples are needed.

Why Gibbs Sampling?

• Compared to other MCMC techniques, Gibbs sampling is:• Easy to implement• Requires little memory• Competitive in speed and performance

Gibbs Sampling for LDA

• The full conditional distribution is:

,

, ,, , ( )

, ,

( | , )d nW dn k n k

d n d n d dn k n

N NP z k z W

N W N K

Probability of Wd,n under topic k

Probability of topic k in document d

,

, ,, , ( )

, ,

1( | , )

d nW dn k n k

d n d n d dn k n

N NP z k z W

Z N W N K

Z = k

Gibbs Sampling for LDA

• Target distribution:

• Initial state of Markov chain: {zn} will have value in {1,2,…,K}.

• Chain run for a number of iterations.

• In each iteration a new state is found by sampling {zn} from

, ,( | , )d n d n dP z k z W

, ,( | , )d n d n dP z k z W

Gibbs Sampling for LDA

• Subsequent samples are taken after appropriate lag to ensure that their autocorrelation is low.

• This is collapsed Gibbs sampling.

• For single sample and are calculated from z.

( )ˆ

d

d

WW kk

k

N

N W

ˆ

dd kk d

N

N K

Fast Collapsed Gibbs Sampling For Latent Dirichlet Allocation

Ian Porteous, David Newman, Alexander Ihler, Arthur Asuncion, Padhraic Smyth, Max Welling

University of California, Irvine

FastLDA: Graphical Representation

FastLDA: Segments

• Sequence of bounds on the Z: Z1,…, Zk

• Z1 Z2 … ZK = Z

• Several slk…sK

k segments for each topic.

• 1st segment: conservative estimate on the probability of the topic given the upper bound Zk on the true normalization factor Z.

• Subsequent segments: corrections for the missing probability mass for a topic given the improved bound.

FastLDA: Segments

Upper Bounds for Z

• Find a sequence of improving bounds on the normalization constant.

• Z defined in terms of component vectors.

• Holder’s inequality to construct initial upper bound.

• Bound intelligently improved for each topic.

Fast LDA Algorithm

• Algorithm:• Sort topics in decreasing order of Nk

d

• u ~ Uniform[0,1]• For topics in order:

• Calculate length of segments.• For each next topic, Zk is improved.• When sum of segments > u:

• Return topic and return.

• Complexity:• Not more than O(K log K) for any operation.

Experiments

• Four large datasets:• NIPS full papers• Enron emails• NY Times news articles• PubMed abstracts

• = 0.01 and = 2/K

• Computations run on workstations with:• Dual Xeon 3.0Ghz processors

• Code compiled by gcc version 3.4.

Results

Speedup : 5-8 times

Results

• Speedup relatively insensitive to number of documents in the corpus.

Results

• Large Dirichlet parameter smooths the distribution of the topics within a document.

• FastLDA needs to visit and compute more topics before drawing a sample.

Discussions

Discussions

• Other domains.

• Other sampling techniques.

• Other distributions other than Dirichlet.

• Parallel computation. • Newman et al. “Scalable parallel topic models”.

• Deciding on the value of K.

• Choices of bounds.

• Reason behind choosing these datasets.

• Are the values mentioned in the paper magic numbers?

• Why were the words having count <10 discarded?

• Assigning weights to words.

Backup Slides

Dirichlet Distribution

• The Dirichlet distribution is an exponential family distribution over the simplex, i.e., positive vectors that sum to one.

• The Dirichlet is conjugate to the multinomial. Given a multinomial observation, the posterior distribution of is a Dirichlet.