Learning Linear, Sparse, Factorial Codes

In previous work (Olshausen & Field 1996), an algorithm was described for learning linear sparse codes which, when trained on natural images, produces a set of basis functions that are spatially localized, oriented, and bandpass (i.e., wavelet-like). This note shows how the algorithm may be interpreted within a maximum-likelihood framework. Several useful insights emerge from this connection: it makes explicit the relation to statistical independence (i.e., factorial coding), it shows a formal relationship to the algorithm of Bell and Sejnowski (1995), and it suggests how to adapt parameters that were previously fixed.

The Logical Problem of Language Change

This paper considers the problem of language change. Linguists must explain not only how languages are learned but also how and why they have evolved along certain trajectories and not others. While the language learning problem has focused on the behavior of individuals and how they acquire a particular grammar from a class of grammars ${cal G}$, here we consider a population of such learners and investigate the emergent, global population characteristics of linguistic communities over several generations. We argue that language change follows logically from specific assumptions about grammatical theories and learning paradigms. In particular, we are able to transform parameterized theories and memoryless acquisition algorithms into grammatical dynamical systems, whose evolution depicts a population's evolving linguistic composition. We investigate the linguistic and computational consequences of this model, showing that the formalization allows one to ask questions about diachronic that one otherwise could not ask, such as the effect of varying initial conditions on the resulting diachronic trajectories. From a more programmatic perspective, we give an example of how the dynamical system model for language change can serve as a way to distinguish among alternative grammatical theories, introducing a formal diachronic adequacy criterion for linguistic theories.

Selecting Relevant Genes with a Spectral Approach

Array technologies have made it possible to record simultaneously the expression pattern of thousands of genes. A fundamental problem in the analysis of gene expression data is the identification of highly relevant genes that either discriminate between phenotypic labels or are important with respect to the cellular process studied in the experiment: for example cell cycle or heat shock in yeast experiments, chemical or genetic perturbations of mammalian cell lines, and genes involved in class discovery for human tumors. In this paper we focus on the task of unsupervised gene selection. The problem of selecting a small subset of genes is particularly challenging as the datasets involved are typically characterized by a very small sample size ?? the order of few tens of tissue samples ??d by a very large feature space as the number of genes tend to be in the high thousands. We propose a model independent approach which scores candidate gene selections using spectral properties of the candidate affinity matrix. The algorithm is very straightforward to implement yet contains a number of remarkable properties which guarantee consistent sparse selections. To illustrate the value of our approach we applied our algorithm on five different datasets. The first consists of time course data from four well studied Hematopoietic cell lines (HL-60, Jurkat, NB4, and U937). The other four datasets include three well studied treatment outcomes (large cell lymphoma, childhood medulloblastomas, breast tumors) and one unpublished dataset (lymph status). We compared our approach both with other unsupervised methods (SOM,PCA,GS) and with supervised methods (SNR,RMB,RFE). The results clearly show that our approach considerably outperforms all the other unsupervised approaches in our study, is competitive with supervised methods and in some case even outperforms supervised approaches.