Metric Learning and Manifolds: Preserving the Intrinsic Geometry

In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover manifold geometry using either local or global features of the data. Building on the Laplacian Eigenmap framework, we propose a new paradigm that offers a guarantee, under reasonable assumptions, that any manifold learning algorithm can be made to preserve the geometry of a data set. Our approach is based on augmenting the output of embedding algorithms with geometric information embodied in the Riemannian metric of the manifold. The Riemannian metric then allows us to define geometric measurements that are faithful to the original data and independent of the algorithm used. In this work, we provide an algorithm for estimating the Riemannian metric from data, consider its consistency, and present an important connection with Gaussian Process Regression and Manifold Regularization.

[@bartholomeosphinx4382]

Whoever filmed this has no understanding whatsoever of the importance of slides. They are shown for a brief second and no more, making the lecture virtually incomprehensible

[@a.bukhari1428]

there is a reason why people created the pause button.

[@NicolauWerneck]

I love these lectures, but it would be great if apart from the title and abstract, the descriptions also had the name of the presenters

[@MrWujie27]

@@NicolauWerneck the name of the presenter is Dominique C. Perrault-Joncas

[@sau002]

Where are the slides?

[@DrKafilatOLiadi]

Good work. Thanks

[@davisidarta]

Dr. Dominique Perrault-Joncas's work is one of the most useful and yet sadly underrated contributions to manifold learning and dimensional reduction. I'm shocked by the low quality of the recording given the gigantic importance of this theme, and the lack of appropriate background by some commenters. As a reference for future viewers, please read his astounding manuscript on metric manifold learning, "Metric Manifold Learning: Preserving the Intrinsic Geometry": sites.stat.washington.edu/mmp/geometry/reading-group17/html/samKslidesRMetric.pdf . Most of the results he exposes are covered in the manuscript.

[@Iamfafafel]

"We believe that there is a more elegant solution to this problem, which is to carry the geometry over along with f instead of trying to correct f itself. Thus, we will take the coordinates f produced by any reasonable embedding algorithm, and augment them with the appropriate (pushforward) metric h that makes (f(M), h) isometric to the original manifold (M, g). We call this procedure metric learning." it sounds like they're solving the problem by defining it away. the whole point finding an embedding to preserve the geometry is to have g agree with the pullback metric on f(M) induced from its embedding into R^{large ambient dimension}. if you do it like how the authors did it, why even bother using an embedding in the first place? i haven't read further details, but this already doesn't look like good news

[@geraldprice3834]

Animated graphics would have been great. What happened, Microsoft? Can't you do that? ;) Missing graphics aside, for anyone minimally versed in this area (such as myself), this is a great lecture that bears replaying over and over.

[@frankd1156]

Wow this is so poor.... shame on Microsoft

[@a_name_a]

Glad I wasn’t the only one to think this.

[@peterbackx393]

this guy uses so much jargon, i wonder about his true depth of knowledge. The lecture was useless for me.

[@davisidarta]

You do need some background to understand state-of-the-art research.

[@alexander_adnan]

this is the worst presentation I ever watched ...