Sparse distance metric learning for embedding compositional data

Zachary D. Kurtz ZACHARY. KURTZ @ MED . NYU . EDU Sackler Institute of Graduate Biomedical Sciences, NYU School of Medicine, New York, NY 10016 Christian L. Müller CMUELLER @ SIMONSFOUNDATION . ORG Richard A. Bonneau RBONNEAU @ SIMONSFOUNDATION . ORG Simons Center for Data Analysis, Simons Foundation, New York, NY 10011

Abstract We propose a novel method for distance metric learning and generalized Aitchison embedding of multi-class data on the (p − 1)-simplex. We consider the p > n setting where p is the number of variables and n the number of data points. This problem setup is motivated by common learning tasks on data sets arising in microbial ecology, where the relative abundance of p species is measured across n environments or patients. Our approach can specifically handle data that contain a large number of zero measurements (zero inflation), a common property for data acquired from targeted high throughput and single-cell sequencing.

to the underlying sparsity of individual samples xk we adopt a novel regularization framework for distance metric learning with the following (squared) Mahalanobis distance (Mateu-Figueras et al., 2013): d2M (xi , xj )

T

=

(a(xi ) − a(xj )) Σ−1 (a(xi ) − a(xj ))

=

∆Tij Q∆ij ,

where Σ denotes the covariance matrix of the Aitchison embedded variables, ∆ij =
(log(xi + bi ) − log(xj + bj )), and Q = PT ΛP the Mahalanobis metric of the Aitchison embedding with Λ the covariance of the log-transformed data.

We are given n data pairs (xk , yk ) where x are pdimensional compositional data vectors that are restricted p−1 to simplex S+ = {x = (x1 , . . . , xi , . . . , xp ): xi ≥ 0, Pthe p i=1 xi = 1}, and each vector xk is associated with a class label yk ∈ {1, . . . , K}. We seek to learn an embedding of the compositional data in Euclidean space under the constraint that distances between xi and xj are small when yi = yj and large for all data points across classes to improve multi-class learning using k-Nearest Neighbor classification (Weinberger & Saul, 2009).

Let X = (x1 , . . . , xi , . . . , xn ) ∈ Rp×n be the given data p×n the matrix of unand B = (b1 , . . . , bi , . . . , bn ) ∈ R+ known pseudocounts. Our novel regularized distance metric learning approach is based on the following non-convex optimization problem:
min − log det Q + tr S T Q

We propose to use Generalized Aitchison Embeddings (GAEs) (Le & Cuturi, 2013) of the form

where S = log(X + B) log(X + B)T and D is the set of dissimilar pairs. The `1 -penalty on the positive definite matrix Q promotes sparsity, which is a common structural assumption in the p > n regime (Friedman √ et al., 2008). Similarly, the nuclear norm kBk∗ = tr BB T promotes low-rank structure in the pseudocount matrix, implying that there is a low-dimensional subspace that can accurately capture the variability of the sample sparsity. The positive scalar quantities s and r are tuning parameters that require data-driven selection.

m

a(x) ≡ P log(x + b) ∈ R , where P ∈ Rm×p , m ≤ p, is a projection matrix and b ∈ Rp+ a positive vector of “pseudocounts". (Le & Cuturi, 2013; 2014) cast the objective of learning the matrix P and vector b as a metric learning problem for the Euclidean distance between two compositions under an Aitchison embedding: da (xi , xj )

= dE (a(xi ), a(xj )) = kP log(xi + b) − P log(xj + b)k2 .

To (i) learn embeddings for high-dimensional compositions when p > n and (ii) adapt the vector of pseudocounts

Q0,B∈Rp×n +

X

dM (xi , xj ) > 1, kQk1 ≤ s, kBk∗ ≤ r

(xi ,xj )∈D

To efficiently solve the proposed optimization problem to local optimality we develop a novel iterative SplitBregman-type algorithm. We show the validity and superior performance of our novel embedding method on a variety of synthetic benchmarks as well as real-world microbial abundance data across different habitats.

Sparse distance metric learning for embedding compositional data

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