Multidimensional Persistence and Noise

A compact three-dimensional four-way vectorial steering module using low-temperature co-fired ceramics multi-layer technology, hybrid-integrated with monolithic microwave integrated circuits, for flexible satellite payloads operating in the Ka-band downlink frequency range of 17–22 GHz is presented. The module features digitally controllable actuators and hermetic sealing at a compact size of 26 mm × 44.5 mm × 4.4 mm. For feeding the module with a single input signal, a one-stage four-way Wilkinson power divider was designed incorporating buried resistors, fabricated, and measured. The divider exhibits a measured return loss of better than 13 dB at the input port and 16 dB at the output ports. The output port isolation exceeds 20 dB, the transmission phase difference remains below 3.1 degrees, and the insertion loss is less than 1.2 dB. The vectorial microwave parameters of the module can be adjusted by 344 degrees for phase and 15.5 dB for amplitude, with 5-bit resolution, and at step sizes of 11.25 degrees and 0.5 dB, respectively. The measured results show that the RMS phase and amplitude errors remain below 3.8 degrees and 0.23 dB.

Topological data analysis is the study of data using techniques from algebraic topology. Often, one begins with a finite set of points representing data and a “filter” function which assigns a real number to each datum. Using both the data and the filter function, one can construct a filtered complex for further analysis. For example, applying the homology functor to the filtered complex produces an algebraic object known as a “one-dimensional persistence module”, which can often be interpreted as a finite set of intervals representing various geometric features in the data. If one runs the above process incorporating multiple filter functions simultaneously, one instead obtains a multidimensional persistence module. Unfortunately, these are much more difficult to interpret. In this article, we analyze the space of multidimensional persistence modules from the perspective of algebraic geometry. We first build a moduli space of a certain subclass of easily analyzed multidimensional persistence modules, which we construct specifically to capture much of the information which can be gained by using multidimensional persistence instead of one-dimensional persistence. We argue that the global sections of this space provide interesting numeric invariants when evaluated against our subclass of multidimensional persistence modules. Finally, we extend these global sections to the space of all multidimensional persistence modules and discuss how the resulting numeric invariants might be used to study data. This paper extends the results of Adcock et al. (Homol Homotopy Appl 18(1), 381–402, 2016) by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson et al. (J Comput Geom 1(1), 72–100, 2010).

We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these modules, and present several examples. We also provide a geometric interpretation for the higher tor-modules of homology modules of multi-filtered simplicial complexes.

<p>TDA mapper is an algorithm used to visualize and analyze big data. TDA mapper is applied to a dataset, X, equipped with a filter function f from X to R. The output of the algorithm is an abstract graph (or simplicial complex). The abstract graph captures topological and geometric information of the underlying space of X.</p><p>One of the interests in TDA mapper is to study whether or not a mapper graph is stable. That is, if a dataset X is perturbed by a small value, and denote the perturbed dataset by X∂, we would like to compare the TDA mapper graph of X to the TDA mapper graph of X∂. Given a topological space X, if the cover of the image of f satisfies certain conditions, Tamal Dey, Facundo Memoli, and Yusu Wang proved that the TDA mapper is stable. That is, the mapper graph of X differs from the mapper graph of X∂ by a small value measured via homology.</p><p>The goal of this thesis is three-fold. The first is to introduce a modified TDA mapper algorithm. The fundamental difference between TDA mapper and the modified version is the modified version avoids the use of filter function. In comparing the mapper graph outputs, the proposed modified mapper is shown to capture more geometric and topological features. We discuss the advantages and disadvantages of the modified mapper.</p><p>Tamal Dey, Facundo Memoli, and Yusu Wang showed that a filtration of covers induce a filtration of simplicial complexes, which in turn induces a filtration of homology groups. While Tamal Dey, Facundo Memoli, and Yusu Wang focused on TDA mapper's application to topological space, the second goal of this thesis is to show DBSCAN clustering gives a filtration of covers when TDA mapper is applied to a point cloud. Hence, DBSCAN gives a filtration of mapper graphs (simplicial complexes) and homology groups. More importantly, DBSCAN gives a filtration of covers, mapper graphs, and homology groups in three parameter directions: bin size, epsilon, and Minpts. Hence, there is a multi-dimensional filtration of covers, mapper graphs, and homology groups. We also note that single-linkage clustering is a special case of DBSCAN clustering, so the results proved to be true when DBSCAN is used are also true when single-linkage is used. However, complete-linkage does not give a filtration of covers in the direction of bin, hence no filtration of simpicial complexes and homology groups exist when complete-linkage is applied to cluster a dataset. In general, the results hold for any clustering algorithm that gives a filtration of covers.</p><p>The third (and last) goal of this thesis is to prove that two multi-dimensional persistence modules (one: with respect to the original dataset, X; two: with respect to the ∂-perturbation of X) are 2∂-interleaved. In other words, the mapper graphs of X and X∂ differ by a small value as measured by homology.</p>