## Archive for the ‘Topological Data Analysis’ Category

### “Almost there….” (Computing Homology)

Friday, April 12th, 2013

We all remember the pilot in Star Wars that kept saying, “Almost there….” Jeremy Kun has us “almost there…” in his latest installment: Computing Homology.

To give you some encouragement, Jeremy concludes the post saying:

The reader may be curious as to why we didn’t come up with a more full-bodied representation of a simplicial complex and write an algorithm which accepts a simplicial complex and computes all of its homology groups. We’ll leave this direct approach as a (potentially long) exercise to the reader, because coming up in this series we are going to do one better. Instead of computing the homology groups of just one simplicial complex using by repeating one algorithm many times, we’re going to compute all the homology groups of a whole family of simplicial complexes in a single bound. This family of simplicial complexes will be constructed from a data set, and so, in grandiose words, we will compute the topological features of data.

If it sounds exciting, that’s because it is! We’ll be exploring a cutting-edge research field known as persistent homology, and we’ll see some of the applications of this theory to data analysis. (bold emphasis added)

Data analysts are needed at all levels.

Do you want to be a spreadsheet data analyst or something a bit harder to find?

### …topological data analysis

Sunday, January 20th, 2013

New big data firm to pioneer topological data analysis by John Burn-Murdoch.

From the post:

A US big data firm is set to establish algebraic topology as the gold standard of data science with the launch of the world’s leading topological data analysis (TDA) platform.

Ayasdi, whose co-founders include renowned mathematics professor Gunnar Carlsson, launched today in Palo Alto, California, having secured $10.25m from investors including Khosla Ventures in the first round of funding. The funds will be used to build on its Insight Discovery platform, the culmination of 12 years of research and development into mathematics, computer science and data visualisation at Stanford. Ayasdi’s work prior to launching as a company has already yielded breakthroughs in the pharmaceuticals industry. In one case it revealed new insights in eight hours – compared to the previous norm of over 100 hours – cutting the turnaround from analysis to clinical trials in the process. The project? CompTop, which I covered here. Does topological data analysis sound more interesting now than before? ### Algebraic Topology and Machine Learning Saturday, August 25th, 2012 Algebraic Topology and Machine Learning – In conjunction with Neural Information Processing Systems (NIPS 2012) September 16, 2012 – Submissions Due October 7, 2012 – Acceptance Notices December 7 or 8 (TBD), 2011, Lake Tahoe, Nevada, USA. From the call for papers: Topological methods and machine learning have long enjoyed fruitful interactions as evidenced by popular algorithms like ISOMAP, LLE and Laplacian Eigenmaps which have been borne out of studying point cloud data through the lens of topology/geometry. More recently several researchers have been attempting to understand the algebraic topological properties of data. Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study and classify topological spaces. The machine learning community thus far has focused almost exclusively on clustering as the main tool for unsupervised data analysis. Clustering however only scratches the surface, and algebraic topological methods aim at extracting much richer topological information from data. The goals of this workshop are: 1. To draw the attention of machine learning researchers to a rich and emerging source of interesting and challenging problems. 2. To identify problems of interest to both topologists and machine learning researchers and areas of potential collaboration. 3. To discuss practical methods for implementing topological data analysis methods. 4. To discuss applications of topological data analysis to scientific problems. We also invite submissions in a variety of areas, at the intersection of algebraic topology and learning, that have witnessed recent activity. Areas of focus for submissions include but are not limited to: 1. Statistical approaches to robust topological inference. 2. Novel applications of topological data analysis to problems in machine learning. 3. Scalable methods for topological data analysis. NIPS2012 site. You will appreciate the “dramatization.” Put on your calendar and/or watch for papers! ### Topological Data Analysis Monday, July 2nd, 2012 Topological Data Analysis by Larry Wasserman. From the post: Topological data analysis (TDA) is a relatively new area of research that spans many disciplines including topology (in particular, homology), statistics, machine learning and computation geometry. The basic idea of TDA is to describe the “shape of the data” by finding clusters, holes, tunnels, etc. Cluster analysis is special case of TDA. I’m not an expert on TDA but I do find it fascinating. I’ll try to give a flavor of what this subject is about. Just in case you want to get in on the ground floor of a new area of research. Larry has citations to the literature in case you need to pick up beach reading. ### Applied topology and Dante: an interview with Robert Ghrist [Sept., 2010] Monday, May 28th, 2012 Applied topology and Dante: an interview with Robert Ghrist by John D. Cook. (September 13, 2010) From the post: Robert Ghrist A few weeks ago I discovered Robert Ghrist via his web site. Robert is a professor of mathematics and electrical engineering. He describes his research as applied topology, something I’d never heard of. (Topology has countless applications to other areas of mathematics, but I’d not heard of much work directly applying topology to practical physical problems.) In addition to his work in applied topology, I was intrigued by Robert’s interest in old books. The following is a lightly-edited transcript of a phone conversation Robert and I had September 9, 2010. If the interview sounds interesting, you may want to read/skim: [2008] R. Ghrist, “Three examples of applied and computational homology,” Nieuw Archief voor Wiskunde 5/9(2). or, [2010] R. Ghrist, “Applied Algebraic Topology & Sensor Networks,” a manu-script text. (caveat! file>50megs!) Applied Topology & Sensor Networks are the notes for an AMS short course. Ghrist recommends continuing with Algebraic Toplogy by Allen Hatcher. (Let me know if you need my shipping address.) Q: Are sensors always mechanical sensors? We speak of them as though that were the case. What if I can’t afford unmanned drones (to say nothing of their pilots) and have$N$people with cellphones? How does a more “discriminating” “sensor” impact the range of capabilities/solutions? ### Inquiry: Algebraic Geometry and Topology Monday, October 31st, 2011 Inquiry: Algebraic Geometry and Topology Speaking of money and such matters, a call for assistance from Quantivity: Algebraic geometry and topology traditionally focused on fairly pure math considerations. With the rise of high-dimensional machine learning, these fields are increasing being pulled into interesting computational applications such as manifold learning. Algebraic statistics and information geometry offer potential to help bridge these fields with modern statistics, especially time-series and random matrices. Early evidence suggests potential for significant intellectual cross-fertilization with finance, both mathematical and computational. Geometrically, richer modeling and analysis of latent geometric structure than available from classic linear algebraic decomposition (e.g. PCA, one of the main workhorses of modern$mathbb{P}\$ finance); for example, cumulant component analysis. Topologically, more effective qualitative analysis of data sampled from manifolds or singular algebraic varieties; for example, persistent homology (see CompTop).

As evidence by Twitter followers, numerous Quantivity readers are experts in these fields. Thus, perhaps the best way to explore is to seek insight from readers.

Readers: please use comments to suggest applied literature from these fields; ideally, although not required, that of potential relevance to finance modeling. All types of literature are requested, from intro texts to survey articles to preprint working papers on specific applications.

These suggestions will be synthesized into one or more subsequent posts, along with appropriate additions to People of Quant Research.

If you or a member of your family knows of any relevant resources, please go to: Inquiry: Algebraic Geometry and Topology and volunteer those resources as comments. You might even make the People of Quant Research list!

I wonder if there would be any interest in tracking bundled instruments using topic maps? That could be an interesting question.