2017

2017-02

  • 2017-02-24
    • Jisu presented his notes [ note ] on the paper by Max Sommerfeld and Axel Munk, “Inference for Empirical Wasserstein Distances on Finite Spaces” [ arXiv ].
  • 2017-02-03

2017-01

  • 2017-01-27
    • Jisu presented the paper by Amy Willis, “Confidence sets for phylogenetic trees” [ arXiv ]. Amy built confidence sets based on central limit theorem, which is from Dennis Barden, Huiling Le, Megan Owen, “Limiting Behaviour of Fréchet Means in the Space of Phylogenetic Trees” [ arXiv ]. The space of phylogenetic tree is constructed in Billera, Holmes, Vogtmann, “Geometry of the Space of Phylogenetic Trees” [ pdf ]. Jisu shared his slides for those two papers for the background [ slides ].
  • 2017-01-20

2016

2016-11

  • 2016-11-03 / 2016-11-18
    • Yen-Chi and Brittany pointed out the paper by Christophe Biscio, Jesper Møller, “The accumulated persistence function, a new useful functional summary statistic for topological data analysis, with a view to brain artery trees and spatial point process applications” [ arXiv ]. The paper seems to essentially propose a variant of our Silhouettes (using a delta-like kernel) instead of a triangle-kernel. So Eric and Brittany are aiming to write a short paper to CCCG that gives an overview of (one-dimensional?) functions derived from the persistence diagrams.
  • 2016-11-12
    • Yen-Chi shared his note on the paper by Sebastian Calonico, Matias D. Cattaneo, Max H. Farrell, “On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference” [ arXiv ], which is about the debiased kernel density estimator [ note ]. They propose to use debiased kernel in KDE so that the resulting KDE has the higher order bias \(O(h^{3})\) and the usual variance \(O\left(\sqrt{\frac{1}{nh^{d}}}\right)\).

2016-10

  • 2016-10-28
    • Larry pointed out the paper by Max Sommerfeld, Axel Munk, “Inference for Empirical Wasserstein Distances on Finite Spaces” [ arXiv ].
  • 2016-10-21 - 2017-01-13
    • topstat meeting is canceled.
  • 2016-10-14
    • Yen-Chi gave a talk about his new paper, “Generalized cluster trees and singular measures” [ pdf ].
  • 2016-10-04
    • Yen-Chi shared a paper by Simon Mak, V. Roshan Joseph, “Support points” [ arXiv ]. The “support points” is similar to coreset, and it might also be related to the k-means clustering.

2016-09

  • 2016-09-23
    • Brittany is visiting Pittsburgh.
    • Jaehyeok gave a talk about how to compute persistent diagram from level set. [ note ]
  • 2016-09-16
    • Jisu presented Stability of Multidimensional Persistent Homology. [ note ]
    • Astrostat talk was about comparing spatial distribution, and he just used correlation.
  • 2016-09-09
    • We skipped meeting.
  • 2016-09-03
    • Larry shared a book by Victor Patrangenaru and Leif Ellingson, “Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis” [ pdf ].
  • 2016-09-02
    • Jisu will present stability theorem on multidimensional persistent homology in two weeks.
    • Jaehyeok will present his work on computing persistent diagram from level set.

2016-08

  • 2016-08-26
    • We discussed some papers to be talked: they are in Projects.
  • 2016-08-25
    • Brittany shared the AIM conference, “Topology of the biomolecular world” [ link ].
  • 2016-08-19
    • We set the meeting time for next semester.
  • 2016-08-17
    • Larry shared a paper by Pratyush Pranav, Herbert Edelsbrunner, Rien van de Weygaert, Gert Vegter, Michael Kerber, Bernard J.T. Jones, Mathijs Wintraecken, “The Topology of the Cosmic Web in Terms of Persistent Betti Numbers” [ arXiv ]. This paper is self-contained, including definition of persistent homology and algorithm for computing it. Their experiments for cosmological simulation is also interesting.
  • 2016-08-12
    • Jessi presented hypothesis test paper. Larry asked how much simulations are done for computing power. Jisu suggested to zoom in and add more test for lower filament proportion in Figure 7, and remove third line in Figure 11 so that there are indices correspondence between rows.

2016-07

  • 2016-07-29
    • Jaehyeok explained his work with Ann about matching galaxy spectra via diffusion map. The objective is to find a common coordinate system to embed two data sets simultaneously and use it to match two data sets. First, diffusion map is applied to embed dataset to euclidean space, and similirarity is adjusted.
  • 2016-07-22
    • Eddie explained his work on how to lower bound the minimax risk of the tangent space estimator \(\mathcal{R}_{n}(\mathcal{P})=\inf_{\hat{T}}\sup_{P\in\mathcal{P}}\mathbb{E}_{P^{n}}\angle(T_{X_{1}}M,\hat{T})\). He considered fixed point design \(\mathcal{R}_{n}(\mathcal{P}_{x})=\inf_{\hat{T}}\sup_{P\in\mathcal{P}}\mathbb{E}_{P^{n}}\angle(T_{x}M,\hat{T})\) first, and then considered randomized design \(\mathcal{R}_{n}(\mathcal{P})=\inf_{\hat{T}}\sup_{P\in\mathcal{P}}\mathbb{E}_{P^{n}}\angle(T_{X_{1}}M,\hat{T})\).
  • 2016-07-05
    • Alessandro shared a paper by Tyrus Berry, Timothy Sauer, “Consistent Manifold Representation for Topological Data Analysis” [ arXiv ]. This paper claims to sharpen persistent homology.

2016-06

  • 2016-06-03
    • Brittany explained what she want to do. She has image data of prostate cancer biopsy images (example). She take the nuclei as a set of points and compute the persistence diagram for the Rips filtration and the DTM filtration. She’d like to have a confidence interval here, but the methods from our annals of stats paper don’t work since the nuclei centers aren’t an iid sample from a distribution. So, instead, she think she can create a bootstrap sample by perturbing the nuclei (perhaps using the radius of the nuclei region as a parameter for how much we can perturb). Doing this, she can create as many ‘new’ data sets that we want, and can compute the distance between the persistence diagrams for the ‘new’ sampled data and the original data set.
    • JaeHyeok and Yen-Chi just discussed about some possible models for this procedure. Seems like there are 3 sources of randomness: number of “nuclei centers”, configuration (locations) of “nuclei centers”, and size distribution of “nuclei”. they think perturbing the sizes only accounts for the 3rd sources–this bootstrap procedure might be valid if we make assumptions to control the randomness about 1 and 2.

2016-05

  • topstat meeting is canceled 2016-05-20 - 2016-07-15.

  • 2016-05-06
    • We discussed test using correlation functions.
    • Added the paper by Ery Arias-Castro, Beatriz Pateiro-López, Alberto Rodríguez-Casal, “Minimax Estimation of the Volume of a Set with Smooth Boundary” [ arXiv ], to Project. The original paper is “Set estimation under convexity type assumptions” [ link ].

2016-04

  • 2016-04-29
    • We discussed how to infer from more general combinatorial structure. For example, distance function is not an average. In particular, let hat{d} be an estimator of distance function d, then a_n(hat{d}-d-b_n) -> W, where W is not necessarily Gaussian. Hence bootstrap is not valid to use.
  • 2016-04-22
    • Jessi presented her work on testing using persistent homology. For two simulation cubes with cold dark matter and warm dark matter, she divided the cubes to subcubes and tested between matched pairs. That seemed to catch difference.
    • We watched Chad Giusti’s webex seminar, Topology and neural computation (video). In the video, Chad Giusti recommended Curto’s paper, “What can topology tell us about the neural code?” (link).
  • 2016-04-15
    • Ann presented her data analysis problem (scribbed note from Jisu). She showed PCA plot of the simulated and real (Brown) galaxy spectra (plot). She referred to Lafon, Keller, Coifman’s paper, “Data Fusion and Multicue Data Matching by Diffusion Maps” (paper). It seems like problem itself is deeply involved with geometry.
  • 2016-04-08
    • Jaehyeok presented his note (note) about Singh, Nikhil, et al.’s paper, “Topological Descriptors of Histology Images” (paper).
    • Jaehyeok also presented his note (note) about Bobrowski et al.’s paper, “Topological consistency via kernel estimation” (paper), and its possible applications to DBSCAN.
    • Yen-Chi presented “Persistent homology analysis of brain artery trees” (paper).
    • Jisu presented his note (note) about Tamal Dey, Jian Sun, Yusu Wang’s paper, “Approximating Cycles in a Shortest Basis of the First Homology Group from Point Data” (paper). We can generalize this in three possible ways: (1) we can try to get representative loop of persistent homology, not the homology, (2) we can relax the manifold support condition, and (3) we can get result for higher dimension.
  • 2016-04-01
    • Brittany shared Dmitry Morozov’s new paper, “Parallel Computation of Persistent Homology using the Blowup Complex” (paper).
    • Larry mentioned that we can work on asymptotic convergence rate of diagram from dtm function that is uniform on parameter m0.
    • Another possible topic is about going back to data and finding representative loops. We can do something like how representative loops statistically behave. Brittany pointed out Tamal Dey, Jian Sun, Yusu Wang’s paper, “Approximating Cycles in a Shortest Basis of the First Homology Group from Point Data” (paper). Brittany also mentioned that smallest loop may not be the best representative loop.
    • Larry pointed out two papers from Steve Marron, “Topological Descriptors of Histology Images” (paper) and “Persistent homology analysis of brain artery trees” (paper).

2016-03

  • 2016-03-25
    • We discussed DBSCAN. The usual density cluster is obtained by taking data points with \(\hat{p}(X_{i})>\lambda\), and taking \(\epsilon\) balls. Cuevas showed that for fixed \(\lambda\), this is consistent. But Alessandro said that it should be done for most of \(\lambda\) simultaneously.
    • Daren presented his note. (note)
  • 2016-03-18
    • We discussed DBSCAN. DBSCAN is a little bit different from density tree. For knn clusters, the consistency for cluster holds, in Theorem 6 from Chaudhuri , Dasgupta, “Rates of convergence for the cluster tree” (paper). Does this kind of consistency holds for DBSCAN as well? We should note that we cannot directly compute cluster of estimated density function. In that sense, Theorem 19 from Balakrishnan, Narayanan, Rinaldo, Singh, Wasserman, “Cluster trees on manifolds” (paper), has missing part (that it considers cluster from estimated density function).
  • 2016-03-11
    • We again discussed “Shape classification based on interpoint distance distributions” paper that we discussed on Nov 6, 2015. (Jisu’s note) (paper) This paper is only considering sets in 2-dim with positive volume, so essentially has theoretical guarantee for 2-dimensional objects. They are assuming uniform distribution, so we can maybe generalize this to arbitrary distribution. Also, Theorem 2 can distinguish only if two sets have different volumes.
    • Brittany suggested that we can have small project (for possibly undergraduate) of building GUI for TDA package.
  • 2016-03-04
    • We skipped topstat meeting.

2016-02

  • 2016-02-26
    • We continued watching webex seminar by Matthew Wright, Visualizing 2-Dimensional Persistent Homology.
  • 2016-02-19
    • Jessi suggested that in hypothesis test, Euler characteristic function seems to lose information, but beats landscapes/silhouettes.
    • Two more projects are added: mode tree and Mapper. See Projects. Larry said that when using kernel density estimator with Gaussian kernel, number of modes always increases as bandwidth goes to 0, so mode tree becomes actually a tree.
    • Jisu gave a brief explanation about multidimensional persistent homology, and suggested to see webex seminar: Matthew Wright, Visualizing 2-Dimensional Persistent Homology.
  • 2016-02-12
    • (slides) Brittany explained Alpha complex and Witness complex.

2016-01

  • 2016-01-29 / 2016-02-05
    • We proposed list of TDA-related projects: see Projects.
  • 2016-01-15
    • (note) Jisu presented results from 2nd School/conference in TDA.

2015

2015-12

  • 2015-12-11 - 2015-12-31
    • Skipping meeting in winter break.
  • 2015-12-04

2015-11

  • 2015-11-27

  • 2015-11-20

  • 2015-11-13

  • 2015-11-06

2015-10

  • 2015-10-30

  • 2015-10-23

  • 2015-10-16

  • 2015-10-09

  • 2015-10-02

2015-09

  • 2015-09-25

  • 2015-09-18

  • 2015-09-11

  • 2015-09-04

2015-05

  • 2015-05-27

2014

2014 Summer

2014-04

  • 2014-03 - 2014-04

2014-03

  • 2014-03-10

  • 2014-03-03

2014-02

  • 2014-02-24

  • 2014-02-17

  • 2014-02-10

  • 2014-02-03

2014-01

  • 2014-01-27

  • 2014-01-20

  • 2014-01-13

2013

2013-12

  • 2013-12-13

  • 2013-12-03

2013-11

  • 2013-11-21

  • 2013-11-14

  • 2013-11-07

2013-10

  • 2013-10-31

  • 2013-10-24

  • 2013-10-17

  • 2013-10-10

  • 2013-10-03

2013-09

  • 2013-09-26