Archive for the ‘Homology’ Category

Homotopy Type Theory

Wednesday, December 4th, 2013

Homotopy Type Theory by Robert Harper. (Course with video lectures, notes, etc.)

Synopsis:

This is a graduate research seminar on Homotopy Type Theory (HoTT), a recent enrichment of Intuitionistic Type Theory (ITT) to include "higher-dimensional" types. The dimensionality of a type refers to the structure of its paths, the constructive witnesses to the equality of pairs of elements of a type, which themselves form a type, the identity type. In general a type is infinite dimensional in the sense that it exhibits non-trivial structure at all dimensions: it has elements, paths between elements, paths between paths, and so on to all finite levels. Moreover, the paths at each level exhibit the algebraic structure of a (higher) groupoid, meaning that there is always the "null path" witnessing reflexivity, the "inverse" path witnessing symmetry, and the "concatenation" of paths witnessing transitivity such that group-like laws hold "up to higher homotopy". Specifically, there are higher-dimensional paths witnessing the associative, unital, and inverse laws for these operations. Altogether this means that a type is a weak ∞-groupoid.

The significance of the higher-dimensional structure of types lies in the concept of a type-indexed family of types. Such families exhibit the structure of a fibration, which means that a path between two indices "lifts" to a transport mapping between the corresponding instances of the family that is, in fact, an equivalence. Thinking of paths as constructive witnesses for equality, this amounts to saying that equal indices give rise to equivalent types, and hence, by univalence, equal elements of the universe in which the family is valued. Thus, for example, if we think of the interval I as a type with two endpoints connected by a path, then an I-indexed family of types must assign equivalent types to the endpoints. In contrast the type B of booleans consists of two disconnected points, so that a B-indexed family of types may assign unrelated types to the two points of B. Similarly, mappings from I into another type A must assign connected points in A to the endpoints of the interval, whereas mappings from B into A are free to assign arbitrary points of A to the two booleans. These preservation principles are central to the structure of HoTT.

In many cases the path structure of a type becomes trivial beyond a certain dimension, called the level of the type. By convention the levels start at -2 and continue through -1, 0, 1, 2, and so on indefinitely. At the lowest, -2, level, the path structure of a type is degenerate in that there is an element to which all other elements are equal; such a type is said to be contractible, and is essentially a singleton. At the next higher level, -1, the type of paths between any two elements is contractible (level -2), which means that any two elements are equal, if there are any elements at all; such as type is a sub-singleton or h-proposition. At the next level, 0, the type of paths between paths between elements is contractible, so that any two elements are equal "in at most one way"; such a type is a set whose types of paths (equality relations) are all h-prop’s. Continuing in this way, types of level 1 are groupoids, those of level 2 are 2-groupoids, and so on for all finite levels.

ITT is capable of expressing only sets, which are types of level 0. Such types may have elements, and two elements may be considered equal in at most one way. A large swath of (constructive) mathematics may be formulated using only sets, and hence is amenable to representation in ITT. Computing applications, among others, require more than just sets. For example, it is often necessary to suppress distinctions among elements of a type so as to avoid over-specification; this is called proof irrelevance. Traditionally ITT has been enriched with an ad hoc treatment of proof irrelevance by introducing a universe of "propositions" with no computational content. In HoTT such propositions are types of level -1, requiring no special treatment or distinction. Such types arise by propositional truncation of a type to render degenerate the path structure of a type above level -1, ensuring that any two elements are equal in the sense of having a path between them.

Propositional truncation is just one example of a higher inductive type, one that is defined by specifying generators not only for its elements, but also for its higher-dimensional paths. The propositional truncation of a type is one that includes all of the elements of the type, and, in addition, a path between any two elements, rendering them equal. It is a limiting case of a quotient type in which only certain paths between elements are introduced, according to whether they are deemed to be related. Higher inductive types also permit the representation of higher-dimensional objects, such as the spheres of arbitrary dimension, as types, simply by specifying their "connectivity" properties. For example, the topological circle consists of a base point and a path starting and ending at that point, and the topological disk may be thought of as two half circles that are connected by a higher path that "fills in" the interior of the circle. Because of their higher path structure, such types are not sets, and neither are constructions such as the product of two circles.

The univalence axiom implies that an equivalence between types (an "isomorphism up to isomorphism") determines a path in a universe containing such types. Since two types can be equivalent in many ways (for example, there can be distinct bijections between two sets), univalence gives rise to types that are not sets, but rather are of a higher level, or dimension. The univalence axiom is mathematically efficient because it allows us to treat equivalent types as equal, and hence interchangeable in all contexts. In informal settings such identifications are often made by convention; in formal homotopy type theory such identifications are true equations.

If you think data types are semantic primitives with universal meaning/understanding, feel free to ignore this posting.

Data types can be usefully treated “as though” they are semantic primitives, but mistaking convenience for truth can be expensive.

The never ending cycle of enterprise level ETL for example. Even when it ends well it is expensive.

And there are all the cases where ETL or data integration don’t end well.

Homotopy Type Theory may not be the answer to those problems, but our current practices are known to not work.

Why not bet on an uncertain success versus the certainty of expense and near-certainty of failure?

Persistent Homology Talk at UIC: Slides

Friday, May 3rd, 2013

Persistent Homology Talk at UIC: Slides by Jeremy Kun.

From the post:

Today I gave a twenty-minute talk at UI Chicago as part of the first annual Chicago Area Student SIAM Conference. My talk was titled “Recent Developments in Persistent Homology,” and it foreshadows the theoretical foundations and computational implementations we’ll be laying out on this blog in the coming months. Here’s the abstract:

Persistent homology is a recently developed technique for analyzing the topology of data sets. We will give a rough overview of the technique and sample successful applications to areas such as natural image analysis & texture classification, breast and liver cancer classification, molecular dynamical systems, and others.

The talk was received very well — mostly, I believe, because I waved my hands on the theoretical aspects and spent most of my time talking about the applications.

The slides.

Jeremy doesn’t hold out much hope the slides will be useful sans the lecture surrounding them.

He does includes references so see what you think of the slides + references.

“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?

Homology Theory — A Primer

Tuesday, April 9th, 2013

Homology Theory — A Primer by Jeremy Kun.

From the post:

This series on topology has been long and hard, but we’re are quickly approaching the topics where we can actually write programs. For this and the next post on homology, the most important background we will need is a solid foundation in linear algebra, specifically in row-reducing matrices (and the interpretation of row-reduction as a change of basis of a linear operator).

Last time we engaged in a whirlwind tour of the fundamental group and homotopy theory. And we mean “whirlwind” as it sounds; it was all over the place in terms of organization. The most important fact that one should take away from that discussion is the idea that we can compute, algebraically, some qualitative features about a topological space related to “n-dimensional holes.” For one-dimensional things, a hole would look like a circle, and for two dimensional things, it would look like a hollow sphere, etc. More importantly, we saw that this algebraic data, which we called the fundamental group, is a topological invariant. That is, if two topological spaces have different fundamental groups, then they are “fundamentally” different under the topological lens (they are not homeomorphic, and not even homotopy equivalent).

Unfortunately the main difficulty of homotopy theory (and part of what makes it so interesting) is that these “holes” interact with each other in elusive and convoluted ways, and the algebra reflects it almost too well. Part of the problem with the fundamental group is that it deftly eludes our domain of interest: we don’t know a general method to compute the damn things!

Jeremy continues his series on topology and promises programs are not far ahead!

Barcodes: The Persistent Topology of Data

Monday, February 27th, 2012

Barcodes: The Persistent Topology of Data by Robert Ghrist.

Abstract:

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets — persistent homology — and a novel representation of this algebraic characterization — barcodes. We sketch an application of these techniques to the classification of natural images.

From the article:

1. The shape of data

When a topologist is asked, “How do you visualize a four-dimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a three-dimensional object?” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer three-dimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations in a strictly lower dimension.

The inference of global structure occurs on much finer scales as well, with regards to converting discrete data into continuous images. Dot-matrix printers, scrolling LED tickers, televisions, and computer displays all communicate images via arrays of discrete points which are integrated into coherent, global objects. This also is a skill we have practiced from childhood. No adult does a dot-to-dot puzzle with anything approaching anticipation.

1.1. Topological data analysis.

Problems of data analysis share many features with these two fundamental integration tasks: (1) how does one infer high dimensional structure from low dimensional representations; and (2) how does one assemble discrete points into global structure.

Now are you interested?

Reminds me of another paper on homology and higher dimensions that I need to finish writing up. Probably not today but later this week.

Found thanks to: Christophe Lalanne’s A bag of tweets / Feb 2012.