The persistence landscape and some of its properties

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

A convenient category of locally preordered spaces

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.Comment: 26 pages, 0 figures, partially presented at GETCO 2005; changes: claim of Prop. 5.11 weakened to finite case and proof changed due to problems with proof of Lemma 3.26, now removed; Eg. 2.7, statement before Lem. 2.11, typos, and other minor problems corrected throughout; extensive rewording; proof of Lem. 3.31, now 3.30, adde

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the $\epsilon$-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $\epsilon$-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $d_I$ satisfies a universality property. This universality result is the central result of the paper. It says that $d_I$ satisfies a stability property generalizing one which $d_B$ is known to satisfy, and that in addition, if $d$ is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $d\leq d_I$. We also show that a variant of this universality result holds for $d_B$, over arbitrary fields. Finally, we show that $d_I$ restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in Foundations of Computational Mathematics. 36 page

Persistent topology for natural data analysis - A survey

Natural data offer a hard challenge to data analysis. One set of tools is being developed by several teams to face this difficult task: Persistent topology. After a brief introduction to this theory, some applications to the analysis and classification of cells, lesions, music pieces, gait, oil and gas reservoirs, cyclones, galaxies, bones, brain connections, languages, handwritten and gestured letters are shown

Topological Machine Learning with Persistence Indicator Functions

Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional scaling, while providing a firm theoretical ground. Many modern machine learning algorithms, however, are based on kernels. This paper presents persistence indicator functions (PIFs), which summarize persistence diagrams, i.e., feature descriptors in topological data analysis. PIFs can be calculated and compared in linear time and have many beneficial properties, such as the availability of a kernel-based similarity measure. We demonstrate their usage in common data analysis scenarios, such as confidence set estimation and classification of complex structured data.Comment: Topology-based Methods in Visualization 201

Design of a factorial experiment with randomization restrictions to assess medical device performance on vascular tissue

Background: Energy-based surgical scalpels are designed to efficiently transect and seal blood vessels using thermal energy to promote protein denaturation and coagulation. Assessment and design improvement of ultrasonic scalpel performance relies on both in vivo and ex vivo testing. The objective of this work was to design and implement a robust, experimental test matrix with randomization restrictions and predictive statistical power, which allowed for identification of those experimental variables that may affect the quality of the seal obtained ex vivo. Methods: The design of the experiment included three factors: temperature (two levels); the type of solution used to perfuse the artery during transection (three types); and artery type (two types) resulting in a total of twelve possible treatment combinations. Burst pressures of porcine carotid and renal arteries sealed ex vivo were assigned as the response variable. Results: The experimental test matrix was designed and carried out as a split-plot experiment in order to assess the contributions of several variables and their interactions while accounting for randomization restrictions present in the experimental setup. The statistical software package SAS was utilized and PROC MIXED was used to account for the randomization restrictions in the split-plot design. The combination of temperature, solution, and vessel type had a statistically significant impact on seal quality. Conclusions: The design and implementation of a split-plot experimental test-matrix provided a mechanism for addressing the existing technical randomization restrictions of ex vivo ultrasonic scalpel performance testing, while preserving the ability to examine the potential effects of independent factors or variables. This method for generating the experimental design and the statistical analyses of the resulting data are adaptable to a wide variety of experimental problems involving large-scale tissue-based studies of medical or experimental device efficacy and performance

Nucleolin binds to a subset of selenoprotein mRNAs and regulates their expression

Selenium, an essential trace element, is incorporated into selenoproteins as selenocysteine (Sec), the 21st amino acid. In order to synthesize selenoproteins, a translational reprogramming event must occur since Sec is encoded by the UGA stop codon. In mammals, the recoding of UGA as Sec depends on the selenocysteine insertion sequence (SECIS) element, a stem-loop structure in the 3′ untranslated region of the transcript. The SECIS acts as a platform for RNA-binding proteins, which mediate or regulate the recoding mechanism. Using UV crosslinking, we identified a 110 kDa protein, which binds with high affinity to SECIS elements from a subset of selenoprotein mRNAs. The crosslinking activity was purified by RNA affinity chromatography and identified as nucleolin by mass spectrometry analysis. In vitro binding assays showed that purified nucleolin discriminates among SECIS elements in the absence of other factors. Based on siRNA experiments, nucleolin is required for the optimal expression of certain selenoproteins. There was a good correlation between the affinity of nucleolin for a SECIS and its effect on selenoprotein expression. As selenoprotein transcript levels and localization did not change in siRNA-treated cells, our results suggest that nucleolin selectively enhances the expression of a subset of selenoproteins at the translational level

Local therapy of cancer with free IL-2

This is a position paper about the therapeutic effects of locally applied free IL-2 in the treatment of cancer. Local therapy: IL-2 therapy of cancer was originally introduced as a systemic therapy. This therapy led to about 20% objective responses. Systemic therapy however was very toxic due to the vascular leakage syndrome. Nevertheless, this treatment was a break-through in cancer immunotherapy and stimulated some interesting questions: Supposing that the mechanism of IL-2 treatment is both proliferation and tumoricidal activity of the tumor infiltrating cells, then locally applied IL-2 should result in a much higher local IL-2 concentration than systemic IL-2 application. Consequently a greater beneficial effect could be expected after local IL-2 application (peritumoral = juxtatumoral, intratumoral, intra-arterial, intracavitary, or intratracheal = inhalation). Free IL-2: Many groups have tried to prepare a more effective IL-2 formulation than free IL-2. Examples are slow release systems, insertion of the IL-2 gene into a tumor cell causing prolonged IL-2 release. However, logistically free IL-2 is much easier to apply; hence we concentrated in this review and in most of our experiments on the use of free IL-2. Local therapy with free IL-2 may be effective against transplanted tumors in experimental animals, and against various spontaneous carcinomas, sarcomas, and melanoma in veterinary and human cancer patients. It may induce rejection of very large, metastasized tumor loads, for instance advanced clinical tumors. The effects of even a single IL-2 application may be impressive. Not each tumor or tumor type is sensitive to local IL-2 application. For instance transplanted EL4 lymphoma or TLX9 lymphoma were not sensitive in our hands. Also the extent of sensitivity differs: In Bovine Ocular Squamous Cell Carcinoma (BOSCC) often a complete regression is obtained, whereas with the Bovine Vulval Papilloma and Carcinoma Complex (BVPCC) mainly stable disease is attained. Analysis of the results of local IL-2 therapy in 288 cases of cancer in human patients shows that there were 27% Complete Regressions (CR), 23% Partial Regressions (PR), 18% Stable Disease (SD), and 32% Progressive Disease (PD). In all tumors analyzed, local IL-2 therapy was more effective than systemic IL-2 treatment. Intratumoral IL-2 applications are more effective than peritumoral application or application at a distant site. Tumor regression induced by intratumoral IL-2 application may be a fast process (requiring about a week) in the case of a highly vascular tumor since IL-2 induces vascular leakage/edema and consequently massive tumor necrosis. The latter then stimulates an immune response. In less vascular tumors or less vascular tumor sites, regression may require 9–20 months; this regression is mainly caused by a cytotoxic leukocyte reaction. Hence the disadvantageous vascular leakage syndrome complicating systemic treatment is however advantageous in local treatment, since local edema may initiate tumor necrosis. Thus the therapeutic effect of local IL-2 treatment is not primarily based on tumor immunity, but tumor immunity seems to be useful as a secondary component of the IL-2 induced local processes. If local IL-2 is combined with surgery, radiotherapy or local chemotherapy the therapeutic effect is usually greater than with either therapy alone. Hence local free IL-2 application can be recommended as an addition to standard treatment protocols. Local treatment with free IL-2 is straightforward and can readily be applied even during surgical interventions. Local IL-2 treatment is usually without serious side effects and besides minor complaints it is generally well supported. Only small quantities of IL-2 are required. Hence the therapy is relatively cheap. A single IL-2 application of 4.5 million U IL-2 costs about 70 Euros. Thus combined local treatment may offer an alternative in those circumstances when more expensive forms of treatment are not available, for instance in resource poor countries