Page 98 - AIH-1-3

Artificial Intelligence in Health ISM: A new multi-view space-learning model

dataset contains the expression of marker genes) and If we multiply each mapping matrix H by H*Q* in
v
redundancy (because the UCI Digits dataset contains Figure 1A, we obtain a representation similar to that in
redundant information in the nature of the images). For Figure 1B. This shows that ISM belongs to the family of
this reason, special emphasis is placed on the analysis of latent space decomposition methods. However, view
these datasets. loadings are a constitutive part of ISM, whereas in other
models, they are derived separately. For example, the
2.2. Methods MOFA+ method uses variance decomposition by factor. 3
2.2.1. Outline of ISM and comparison with other latent (d) Important implications of ISM’s preliminary
space approaches embedding
Before delving into the details of the ISM workflow, we As will be detailed in the workflow description, ISM
present the main underlying ideas with an illustrative begins by applying NMF to the concatenated views.
figure (Figure 1A) and compare ISM with other latent Importantly, NMF can be applied to each view X separately,
space approaches (Figure 1B). The different views are v nmf nmfT
represented by heatmaps on the left side of both panels, leading to view-specific decompositions X = W v H v
v
with attributes on the vertical axis and observations on the before ISM itself is applied to the m NMF-transformed
horizontal axis. views W v nmf . In this case, the view mapping returned by
(a) ISM ISM, H ism , refers to the NMF components of each W v nmf .
v
However, by embedding the W nmf in a 3D array, ISM
In the central part of Figure 1A, each non-negative v
view Xv is decomposed into the product of two non- allows H ism to be mapped back to the original views
v
negative matrices, H and W , using NMF. Each W matrix through simple chained matrix multiplication such that:
v
v
v
corresponds to the transformation of a particular view v X = W H T with H = H nmf HH Q . We refer to this
*
*
ism
*
to a latent space common to all transformed views. ISM v v v v v v
ensures that the transformed views, W , share the same alternative approach as integrated latent source model
v
number and type of latent attributes, as explained in the (ILSM). As shown in the results (Section 3) and discussion
detailed description. This transforming process, which (Section 4) sections, ILSM offers important advantages in
we call embedding, results in a 3D array, or tensor. The several respects.
corresponding H matrices contain the loadings of the 2.2.2. Compared methods
v
original attributes on each component. We refer to
these matrices as the mapping between the original and In this article, we compare ISM and ILSM with multi-view
3,4
2,14
transformed views. multidimensional scaling (MVMDS), MOFA+, group
factor analysis (GFA), and Multi-Omics Wasserstein
18
In the right part of Figure 1A, the 3D array is inteGrative anaLysIs (MOWGLI). Below is a brief
21
decomposed into the tensor product of three matrices: W*, description of each of these methods.
H*, and Q* using NTF. W* contains the meta-scores – the (a) MVMDS: After computing and double-centering
single transformation to the latent space common to all the Euclidean distance matrices for each of the
views. H* and Q* contain the loadings of the latent views, MVMDS estimates the common principal
attributes and views, respectively, on each NTF component. components of the matrices in a manner similar to the
Each row of Q* is represented by a diagonal matrix, where generalization of principal component analysis (PCA)
the diagonal contains the loadings for a particular view. for multiple covariance matrices
This allows for each view of the tensor to translate the (b) MOFA+ and GFA: Both models are formulated in
tensor product into a simple matrix product WH Q( * * v ) , a probabilistic Bayesian framework, where prior
*
T
as seen in Figure 1A. distributions are placed on all unobserved variables of
(b) Other latent space approaches the model, using a standard normal prior for the factors
W and sparsity priors for the mapping matrices H
v
In the right part of Figure 1B, each view v is decomposed (c) MOWGLI: This model is a multi-view generalization
into the product of two matrices, H and W, using the of NMF, using optimal transport instead of the
v
latent space method algorithm. As with ISM, W contains Frobenius cost function and regularization parameters
the meta-scores – the single transformation in the latent that ensure sparsity and consistency between model
space common to all views. parameters across different views. A sum-to-one
(c) Comparison between ISM and other latent space constraint is applied to the common factors W to give
approaches them a probabilistic interpretation.
Volume 1 Issue 3 (2024) 92 doi: 10.36922/aih.3427