Sequential Learning of Principal Curves: Summarizing Data Streams on the Fly

When confronted with massive data streams, summarizing data with dimension reduction methods such as PCA raises theoretical and algorithmic pitfalls. A principal curve acts as a nonlinear generalization of PCA, and the present paper proposes a novel algorithm to automatically and sequentially learn principal curves from data streams. We show that our procedure is supported by regret bounds with optimal sublinear remainder terms. A greedy local search implementation (called slpc, for sequential learning principal curves) that incorporates both sleeping experts and multi-armed bandit ingredients is presented, along with its regret computation and performance on synthetic and real-life data.

Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

Principal curve approaches for inferring 3D chromatin architecture

Summary Three-dimensional (3D) genome spatial organization is critical for numerous cellular processes, including transcription, while certain conformation-driven structural alterations are frequently oncogenic. Genome architecture had been notoriously difficult to elucidate, but the advent of the suite of chromatin conformation capture assays, notably Hi-C, has transformed understanding of chromatin structure and provided downstream biological insights. Although many findings have flowed from direct analysis of the pairwise proximity data produced by these assays, there is added value in generating corresponding 3D reconstructions deriving from superposing genomic features on the reconstruction. Accordingly, many methods for inferring 3D architecture from proximity data have been advanced. However, none of these approaches exploit the fact that single chromosome solutions constitute a one-dimensional (1D) curve in 3D. Rather, this aspect has either been addressed by imposition of constraints, which is both computationally burdensome and cell type specific, or ignored with contiguity imposed after the fact. Here, we target finding a 1D curve by extending principal curve methodology to the metric scaling problem. We illustrate how this approach yields a sequence of candidate solutions, indexed by an underlying smoothness or degrees-of-freedom parameter, and propose methods for selection from this sequence. We apply the methodology to Hi-C data obtained on IMR90 cells and so are positioned to evaluate reconstruction accuracy by referencing orthogonal imaging data. The results indicate the utility and reproducibility of our principal curve approach in the face of underlying structural variation.

Principal curve approaches for inferring 3D chromatin architecture

Three dimensional (3D) genome spatial organization is critical for numerous cellular processes, including transcription, while certain conformation-driven structural alterations are frequently oncogenic. Genome architecture had been notoriously difficult to elucidate, but the advent of the suite of chromatin conformation capture assays, notably Hi-C, has transformed understanding of chromatin structure and provided downstream biological insights. Although many findings have flowed from direct analysis of the pairwise proximity data produced by these assays, there is added value in generating corresponding 3D reconstructions deriving from superposing genomic features on the reconstruction. Accordingly, many methods for inferring 3D architecture from proximity d hyperrefata have been advanced. However, none of these approaches exploit the fact that single chromosome solutions constitute a one dimensional (1D) curve in 3D. Rather, this aspect has either been addressed by imposition of constraints, which is both computationally burdensome and cell type specific, or ignored with contiguity imposed after the fact. Here we target finding a 1D curve by extending principal curve methodology to the metric scaling problem. We illustrate how this approach yields a sequence of candidate solutions, indexed by an underlying smoothness or degrees-of-freedom parameter, and propose methods for selection from this sequence. We apply the methodology to Hi-C data obtained on IMR90 cells and so are positioned to evaluate reconstruction accuracy by referencing orthogonal imaging data. The results indicate the utility and reproducibility of our principal curve approach in the face of underlying structural variation.

Human immune system variation during one year

SUMMARYThe human immune system varies extensively between individuals, but variation within individuals over time has not been well characterized. Systems-level analyses allow for simultaneous quantification of many interacting immune system components, and the inference of global regulatory principles. Here we present a longitudinal, systems-level analysis in 99 healthy adults, 50 to 65 years of age and sampled every 3rd month during one year. We describe the structure of inter-individual variation and characterize extreme phenotypes along a principal curve. From coordinated measurement fluctuations, we infer relationships between 115 immune cell populations and 750 plasma proteins constituting the blood immune system. While most individuals have stable immune systems, the degree of longitudinal variability is an individual feature. The most variable individuals, in the absence of overt infections, exhibited markers of poor metabolic health suggestive of a functional link between metabolic and immunologic homeostatic regulation.HIGHLIGHTSLongitudinal variation in immune cell composition during one yearInter-individual variation can be described along a principal curveImmune cell and protein relationships are inferredVariability over time is an individual feature correlating with markers of poor metabolic health

A Length Penalized Probabilistic Principal Curve Algorithm With Applications To Handwritten Digits And Pharmacologic Colon Imaging

The classical Principal Curve algorithm was developed as a nonlinear version of principal component analysis to model curves. However, existing principal curve algorithms with classical penalties, such as smoothness or ridge penalties, lack the ability to deal with complex curve shapes. In this manuscript, we introduce a robust and stable length penalty which solves issues of unnecessary curve complexity, such as the self-looping, that arise widely in principal curve algorithms. A novel probabilistic mixture regression model is formulated. A modified penalized EM(Expectation Maximization) Algorithm was applied to the model to obtain the penalized MLE. Two applications of the algorithm were performed. In the first, the algorithm was applied to the MNIST dataset of handwritten digits to find the centerline, not unlike defining a TrueType font. We demonstrate that the centerline can be recovered with this algorithm. In the second application, the algorithm was applied to construct a three dimensional centerline through single photon emission computed tomography images of the colon arising from the study of pre-exposure prophylaxis for HIV. The centerline in this application is crucial for understanding the distribution of the antiviral agents in the colon for HIV prevention. The new algorithms improves on previous applications of principal curves to this data.