Multi-View Spectral Clustering Based on Multi-Smooth Representation Fusion for Cancer Subtype Prediction

It is a vital task to design an integrated machine learning model to discover cancer subtypes and understand the heterogeneity of cancer based on multiple omics data. In recent years, some multi-view clustering algorithms have been proposed and applied to the prediction of cancer subtypes. Among them, the multi-view clustering methods based on graph learning are widely concerned. These multi-view approaches usually have one or more of the following problems. Many multi-view algorithms use the original omics data matrix to construct the similarity matrix and ignore the learning of the similarity matrix. They separate the data clustering process from the graph learning process, resulting in a highly dependent clustering performance on the predefined graph. In the process of graph fusion, these methods simply take the average value of the affinity graph of multiple views to represent the result of the fusion graph, and the rich heterogeneous information is not fully utilized. To solve the above problems, in this paper, a Multi-view Spectral Clustering Based on Multi-smooth Representation Fusion (MRF-MSC) method was proposed. Firstly, MRF-MSC constructs a smooth representation for each data type, which can be viewed as a sample (patient) similarity matrix. The smooth representation can explicitly enhance the grouping effect. Secondly, MRF-MSC integrates the smooth representation of multiple omics data to form a similarity matrix containing all biological data information through graph fusion. In addition, MRF-MSC adaptively gives weight factors to the smooth regularization representation of each omics data by using the self-weighting method. Finally, MRF-MSC imposes constrained Laplacian rank on the fusion similarity matrix to get a better cluster structure. The above problems can be transformed into spectral clustering for solving, and the clustering results can be obtained. MRF-MSC unifies the above process of graph construction, graph fusion and spectral clustering under one framework, which can learn better data representation and high-quality graphs, so as to achieve better clustering effect. In the experiment, MRF-MSC obtained good experimental results on the TCGA cancer data sets.

Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

Homological branching law for $$({\mathrm {GL}}_{n+1}(F), {\mathrm {GL}}_n(F))$$: projectivity and indecomposability

Let F be a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from $${\mathrm {GL}}_{n+1}(F)$$ GL n + 1 ( F ) to $${\mathrm {GL}}_n(F)$$ GL n ( F ) . A main result shows that each Bernstein component of an irreducible smooth representation of $${\mathrm {GL}}_{n+1}(F)$$ GL n + 1 ( F ) restricted to $${\mathrm {GL}}_n(F)$$ GL n ( F ) is indecomposable. We also classify all irreducible representations which are projective when restricting from $${\mathrm {GL}}_{n+1}(F)$$ GL n + 1 ( F ) to $${\mathrm {GL}}_n(F)$$ GL n ( F ) . A main tool of our study is a notion of left and right derivatives, extending some previous work joint with Gordan Savin. As a by-product, we also determine the branching law in the opposite direction.