Archive for the ‘Geometry’ Category

Visualizing What Your Computer (and Science) Ignore (mostly)

Thursday, November 12th, 2015

Deviation Magnification: Revealing Departures from Ideal Geometries by Neal Wadhwa, Tali Dekel, Donglai Wei, Frédo Durand, William T. Freeman.

Abstract:

Structures and objects are often supposed to have idealized geome- tries such as straight lines or circles. Although not always visible to the naked eye, in reality, these objects deviate from their idealized models. Our goal is to reveal and visualize such subtle geometric deviations, which can contain useful, surprising information about our world. Our framework, termed Deviation Magnification, takes a still image as input, fits parametric models to objects of interest, computes the geometric deviations, and renders an output image in which the departures from ideal geometries are exaggerated. We demonstrate the correctness and usefulness of our method through quantitative evaluation on a synthetic dataset and by application to challenging natural images.

The video for the paper is quite compelling:

Read the full paper here: http://people.csail.mit.edu/nwadhwa/deviation-magnification/DeviationMagnification.pdf

From the introduction to the paper:

Many phenomena are characterized by an idealized geometry. For example, in ideal conditions, a soap bubble will appear to be a perfect circle due to surface tension, buildings will be straight and planetary rings will form perfect elliptical orbits. In reality, however, such flawless behavior hardly exists, and even when invisible to the naked eye, objects depart from their idealized models. In the presence of gravity, the bubble may be slightly oval, the building may start to sag or tilt, and the rings may have slight perturbations due to interactions with nearby moons. We present Deviation Magnification, a tool to estimate and visualize such subtle geometric deviations, given only a single image as input. The output of our algorithm is a new image in which the deviations from ideal are magnified. Our algorithm can be used to reveal interesting and important information about the objects in the scene and their interaction with the environment. Figure 1 shows two independently processed images of the same house, in which our method automatically reveals the sagging of the house’s roof, by estimating its departure from a straight line.

Departures from “idealized geometry” make for captivating videos but there is a more subtle point that Deviation Magnification will help bring to the fore.

“Idealized geometry,” just like discrete metrics for attitude measurement or metrics of meaning, etc. are all myths. Useful myths as houses don’t (usually) fall down, marketing campaigns have a high degree of success, and engineering successfully relies on approximations that depart from the “real world.”

Science and computers have a degree of precision that has no counterpart in the “real world.”

Watch the video again if you doubt that last statement.

Whether you are using science and/or a computer, always remember that your results are approximations based upon approximations.

I first saw this in Four Short Links: 12 November 2015 by Nat Torkington.

GSI2013 – Geometric Science of Information

Tuesday, June 4th, 2013

GSI2013 – Geometric Science of Information 28-08-2013 – 30-08-2013 (Paris) (program detail)

Abstracts.

From the homepage:

The objective of this SEE Conference hosted by MINES ParisTech, is to bring together pure/applied mathematicians and engineers, with common interest for Geometric tools and their applications for Information analysis, with active participation of young researchers for deliberating emerging areas of collaborative research on “Information Geometry Manifolds and Their Advanced Applications”.

I first saw this at: Geometric Science of Information (GSI): programme is out!

Visual Computing: Geometry, Graphics, and Vision (source code)

Friday, April 12th, 2013

Frank Nielsen blogged today that he had posted the C++ source code for “Visual Computing: Geometry, Graphics, and Vision.”

See: Source codes for all chapters of “Visual Computing: Geometry, Graphics, and Vision”

New demos are reported to be on the way.

Esri Geometry API

Wednesday, March 27th, 2013

Esri Geometry API

From the webpage:

geometry-api-java

The Esri Geometry API for Java can be used to enable spatial data processing in 3rd-party data-processing solutions. Developers of custom MapReduce-based applications for Hadoop can use this API for spatial processing of data in the Hadoop system. The API is also used by the Hive UDF’s and could be used by developers building geometry functions for 3rd-party applications such as Cassandra, HBase, Storm and many other Java-based “big data” applications.

Features

  • API methods to create simple geometries directly with the API, or by importing from supported formats: JSON, WKT, and Shape
  • API methods for spatial operations: union, difference, intersect, clip, cut, and buffer
  • API methods for topological relationship tests: equals, within, contains, crosses, and touches

This looks particularly useful for mapping the rash of “public” data sets to facts on the ground.

Particularly if income levels, ethnicity, race, religion and other factors are taken into account.

Might give more bite to the “excess population,” aka the “47%” people speak so casually about.

Additional resources:

ArcGIS Geodata Resource Center

ArcGIS Blog

twitter@esri

Computational Information Geometry

Sunday, January 27th, 2013

Computational Information Geometry by Frank Nielsen.

From the homepage:

Computational information geometry deals with the study and design of efficient algorithms in information spaces using the language of geometry (such as invariance, distance, projection, ball, etc). Historically, the field was pioneered by C.R. Rao in 1945 who proposed to use the Fisher information metric as the Riemannian metric. This seminal work gave birth to the geometrization of statistics (eg, statistical curvature and second-order efficiency). In statistics, invariance (by non-singular 1-to-1 reparametrization and sufficient statistics) yield the class of f-divergences, including the celebrated Kullback-Leibler divergence. The differential geometry of f-divergences can be analyzed using dual alpha-connections. Common algorithms in machine learning (such as clustering, expectation-maximization, statistical estimating, regression, independent component analysis, boosting, etc) can be revisited and further explored using those concepts. Nowadays, the framework of computational information geometry opens up novel horizons in music, multimedia, radar, and finance/economy.

Numerous resources including publications, links to conference proceedings (some with videos), software and other materials, including a tri-lingual dictionary, Japanese, English, French, of terms in information geometry.

Dictionary of computational information geometry

Sunday, January 27th, 2013

Dictionary of computational information geometry (PDF) by Frank Nielsen. (Compiled January 23, 2013)

The title is a bit misleading.

It should read: “[Tri-Lingual] Dictionary of computational information geometry.”

Terms are defined in:

Japanese-English

English-Japanese

Japanese-French

An excellent resource in a linguistically diverse world!

Make your own buckyball

Wednesday, October 31st, 2012

Make your own buckyball by John D. Cook.

From the post:

This weekend a couple of my daughters and I put together a buckyball from a Zometool kit. The shape is named for Buckminster Fuller of geodesic dome fame. Two years after Fuller’s death, scientists discovered that the shape appears naturally in the form of a C60 molecule, named Buckminsterfullerene in his honor. In geometric lingo, the shape is a truncated icosahedron. It’s also the shape of many soccer balls.

Don’t be embarrassed to use these at the office.

According to the PR, Roger Penrose does.

Geometric properties of graph layouts optimized for greedy navigation

Saturday, August 4th, 2012

Geometric properties of graph layouts optimized for greedy navigation by Sang Hoon Lee and Petter Holme.

The graph layouts used for complex network studies have been mainly been developed to improve visualization. If we interpret the layouts in metric spaces such as Euclidean ones, however, the embedded spatial information can be a valuable cue for various purposes. In this work, we focus on the navigational properties of spatial graphs. We use an recently user-centric navigation protocol to explore spatial layouts of complex networks that are optimal for navigation. These layouts are generated with a simple simulated annealing optimization technique. We compared these layouts to others targeted at better visualization. We discuss the spatial statistical properties of the optimized layouts for better navigability and its implication.

Despite my misgivings about metric spaces, to say nothing of Euclidean ones, for some data, this looks particularly useful.

If you had the optimal layout for navigation of a graph, how would you recognize it? Aside from voicing your preference or choice?

Difficult question but one that the authors are pursuing.

It may be that measurement of “navigability” is possible.

Even if we have to accept that hidden factors are behind the “navigability” measurement.

Conferences on Intelligent Computer Mathematics (CICM 2012)

Saturday, July 14th, 2012

Conferences on Intelligent Computer Mathematics (CICM 2012) (talks listing)

From the “general information” page:

As computers and communications technology advance, greater opportunities arise for intelligent mathematical computation. While computer algebra, automated deduction, mathematical publishing and novel user interfaces individually have long and successful histories, we are now seeing increasing opportunities for synergy among these areas.

The conference is organized by Serge Autexier (DFKI) and Michael Kohlhase (JUB), takes place at Jacobs University in Bremen and consists of five tracks

The overall programme is organized by the General Program Chair Johan Jeuring.

Which I located by following the conference reference in: An XML-Format for Conjectures in Geometry (Work-in-Progress)

A real treasure trove of research on searching, semantics, integration, focused on computers and mathematics.

Expect to see citations to work reported here and in other CICM proceedings.

An XML-Format for Conjectures in Geometry (Work-in-Progress)

Saturday, July 14th, 2012

An XML-Format for Conjectures in Geometry (Work-in-Progress) by Pedro Quaresma.

Abstract:

With a large number of software tools dedicated to the visualisation and/or demonstration of properties of geometric constructions and also with the emerging of repositories of geometric constructions, there is a strong need of linking them, and making them and their corpora, widely usable. A common setting for interoperable interactive geometry was already proposed, the i2g format, but, in this format, the conjectures and proofs counterparts are missing. A common format capable of linking all the tools in the field of geometry is missing. In this paper an extension of the i2g format is proposed, this extension is capable of describing not only the geometric constructions but also the geometric conjectures. The integration of this format into the Web-based GeoThms, TGTP and Web Geometry Laboratory systems is also discussed.

The author notes open questions as:

  • The xml format must be complemented with an extensive set of converters allowing the exchange of information between as many geometric tools as possible.
  • The databases queries, as in TGTP, raise the question of selecting appropriate keywords. A fine grain index and/or an appropriate geometry ontology should be addressed.
  • The i2gatp format does not address proofs. Should we try to create such a format? The GATPs produce proofs in quite different formats, maybe the construction of such unifying format it is not possible and/or desirable in this area.

The “keywords,” “fine grained index,” “geometry ontology,” question yells “topic map” to me.

You?

PS: Converters and different formats also say “topic map,” just not as loudly to me. Your volume may vary. (YVMV)

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Geometric and Quantum Methods for Information Retrieval

Tuesday, June 5th, 2012

Geometric and Quantum Methods for Information Retrieval by Yaoyong Li and Hamish Cunningham.

Abstract:

This paper reviews the recent developments in applying geometric and quantum mechanics methods for information retrieval and natural language processing. It discusses the interesting analogies between components of information retrieval and quantum mechanics. It then describes some quantum mechanics phenomena found in the conventional data analysis and in the psychological experiments for word association. It also presents the applications of the concepts and methods in quantum mechanics such as quantum logic and tensor product to document retrieval and meaning of composite words, respectively. The purpose of the paper is to give the state of the art on and to draw attention of the IR community to the geometric and quantum methods and their potential applications in IR and NLP.

More complex models can (may?) lead to better IR methods, but:

Moreover, as Hilbert space is the mathematical foundation for quantum mechanics (QM), basing IR on Hilbert space creates an analogy between IR and QM and may usefully bring some concepts and methods from QM into IR. (p.24)

is a dubious claim at best.

The “analogy” between QM and IR makes the point:

QM IR
a quantum system a collection of object for retrieval
complex Hilbert space information space
state vector objects in collection
observable query
measurement search
eigenvalues relevant or not for one object
probability of getting one eigenvalue relevance degree of object to query

The authors are comparing apples and oranges. For example, “complex Hilbert space” and “information space.”

A “complex Hilbert space” is a model that has been found useful with another model, one called quantum mechanics.

An “information space,” on the other hand, encompasses models known to use “complex Hilbert spaces” and more. Depends on the information space of interest.

Or the notion of “observable” being paired with “query.”

Complex Hilbert spaces may be quite useful for IR, but tying IR to quantum mechanics isn’t required to make use of it.

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.