Workshop Schedule

Wednesday June 25, 2014

Talk Abstracts, Slides, and Papers

Persistent homology and the structure of data.
Mikael Vejdemo-Johansson (KTH Royal Institute of Technology, Sweden)

In Topological Data Analysis and novel topological approaches to machine learning, one core technique is Persistent Homology. In this tutorial we will go through the fundamental definitions and their motivations, and look at concrete examples and computational strategies for computing persistent homology of datasets and using the results to extract shape information that goes beyond clustering of data to provide topologically based information.

Persistent Homology for Characterizing Stimuli Response in the Primary Visual Cortex.
[paper pdf]

Avani Wildani and Tatyana O. Sharpee (SALK)
The neural code is one of the largest and most perplexing frontiers in modern science. Much of the difficulty of studying networks in the brain lies in the fact that biological datasets are often not directly comparable, restricting most analyses to a very small sample size. We use persistent homology to investigate a basic classification problem in the primary visual cortex (V1): how do we differentiate neural responses to natural scenes from neural response to a Gaussian noise or other synthetic image? Topological techniques are particularly suited to stimuli characterization because they are coordinate free, allowing for comparison across brains, and resilient to deformation, decreasing the influence of noise. We show promising results towards obtaining a classification based on the first-order Betti number of the V1 neural response.

Stability and statistical properties of topological information inferred from data.
Fred Chazal (INRIA)

Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. It is usually computed from filtrations built on top of data sets sampled from some unknown (metric) space, providing "topological signatures" revealing the structure of the underlying space. In this talk we will present a few stability and statistical properties of persistence diagrams that allow to efficiently compute  relevant topological signatures that are robust to the presence of outliers in the data.

Title: Using Topology to get a Global View of Machine Learning Problems
Anthony Bak (Stanford and Ayasdi)
[pdf (34MB)]

We will discuss the point of view being developed at Ayasdi on how to use Topological Data Analysis (TDA) in concert with traditional statistical and machine learning methods to build better more accurate models.

Applied Hodge Theory.
Yuan Yao (Peking University)

Hodge Theory is a milestone bridging differential geometry and algebraic topology. It studies certain functions (called forms) on data rather than data points themselves, and brings an optimization perspective to decompose such functions adaptive to the underlying topology. Recently Hodge Theory inspires rising applications in computer vision, multimedia, statistical ranking, and game theory, in addition to traditional applications in mechanics etc. In this tutorial we give an introduction to Hodge Theory with examples in these applications.

Parametric Inference using Persistence Diagrams: A Case Study in Population Genetics

Kevin Emmett, Daniel Rosenbloom, Pablo Camara, Raul Rabadan
Persistent homology computes topological invariants from point cloud data. Recent work has focused on developing statistical methods for data analysis in this framework. We show that, in certain models, parametric inference can be performed using statistics defined on the computed invariants. We develop this idea with a model from population genetics, the coalescent with recombination. We apply our model to an influenza dataset, identifying two scales of topological structure which have a distinct biological interpretation.

Towards an Efficient Discovery of Topological Representative Subgraphs
Wajdi Dhifli
The combinatorial nature of graphs makes performing exact or approximate isomorphism very costly. In this paper, we propose an approach that mines a subset of topological representative subgraphs among frequent ones. The proposed approach overcomes the costly exact or approximate isomorphism by measuring the overall structural similarity based on a considered set of topological attributes. It also allows detecting hidden structural similarities (density, diameter, etc.) that existing approaches ignore. In addition, the proposed approach is flexible and can be easily extended with other attributes depending on the application. Empirical studies on real and synthetic graph datasets show the efficiency of our approach.

Minimum Curvilinearity
Carlo Vittorio Cannistraci

Data-based Manifold Reconstruction via Tangent Bundle Manifold Learning

Alexander Bernstein and Alexander Kuleshov
The goal of Manifold Learning (ML) is to find a description of low-dimensional structure of an unknown q-dimensional manifold embedded in high-dimensional ambient Euclidean space R^p, q < p, from their finite samples. There are variety of formulations of the problem. The methods of Manifold Approximation (MA) reconstruct (estimate) the manifold but don’t find a low-dimensional parameterization on the manifold. The most of Manifold Embedding (ME) methods find the low-dimensional parameterization but don’t reconstruct the manifold. In the paper, the ML is considered as Tangent Bundle Manifold Learning (TBML) in which the manifold, its tangent spaces and low-dimensional representation accurately reconstructed from the sample. A new geometrically motivated method for the TBML is presented, which also gives a new solution for the MA and ME.

Algebraic and topological perspectives on semisupervised learning.
Mikael Vejdemo-Johansson

By describing algebraic structures representing geometric and topological features of dataset with generators and relations, we can use the entire force of contemporary computational commutative algebra to extend topological data analysis techniques. In particular, we are investigating the power of working with maps between finitely presented modules as a way to represent prior knowledge in topological structure descriptions of point cloud data. We will present an approach to semisupervised classification that introduces additional algebraic relations to represent known partial classifications and thereby provide both a guide to the classification process and a guide to parameter choices for classification with topological techniques. This is joint work with Primoz Skraba.

The global-first topological definition of perceptual objects.
Lin Chen (CAS)
What is a perceptual object? Intuitively, it is the holistic identity preserved over shape-changing transformations. According to the global-first topological approach, this core intuitive notion of an object can be characterized as topological invariants, such as connectivity and holes. Behavioral experiments demonstrated that changes in topological properties disturbed object continuity, leading to the perception of an emergence of a new object; and fMRI experiments showed that the topological changes activated the anterior temporal lobe and amygdale.