Topological Data Analysis for Object Data

Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes to vectorize object data and perform statistical analysis. We apply to this pipeline to some biological images that were previously shown to be challenging to study using shape theory. Surprisingly, the most persistent features are shown to be "topological noise" and the statistical analysis depends on the less persistent features which we refer to as the "geometric signal". We also describe the first steps to a new approach to using topology for object data analysis, which applies topology to distributions on object spaces.Comment: 16 pages, 12 figure

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

Somatic gene therapy for cancer. The utility of transferrinfection in generating ‘tumor vaccines’

The last few years have seen the development of a branch of somatic gene therapy which aims at strengthening the immune surveillance of the body, leading to eradication of disseminated cancer tumor cells and occult micrometastases after surgical removal of the primary tumor. Such a tumor vaccination protocol calls for cultivation of the primary tumor tissue and the insertion of one of three types of genes into the isolated cultured tumor cells followed by irradiation of the transfected or transduced cells to render them incapable of further proliferation. The cells so treated constitute the ‘tumor vaccine’. A review of the literature suggests that for mouse models, in the initial period after inoculation, rejection of the tumor cells is usually effected by non-T-cell immunity, whereas the long-term systemic immune response is based on cytotoxic T-cells. High expression of the gene inserted into the tumor cells may be critical for the success of the vaccination procedure. Examples are given which indicate that transferrinfection, a procedure to introduce genes by adenovirus-augmented receptor-mediated endocytosis, meets some important prerequisites for successful application of this type of gene therapy

Quantifying similarity of pore-geometry in nanoporous materials

In most applications of nanoporous materials the pore structure is as important as the chemical composition as a determinant of performance. For example, one can alter performance in applications like carbon capture or methane storage by orders of magnitude by only modifying the pore structure. For these applications it is therefore important to identify the optimal pore geometry and use this information to find similar materials. However, the mathematical language and tools to identify materials with similar pore structures, but different composition, has been lacking. We develop a pore recognition approach to quantify similarity of pore structures and classify them using topological data analysis. This allows us to identify materials with similar pore geometries, and to screen for materials that are similar to given top-performing structures. Using methane storage as a case study, we also show that materials can be divided into topologically distinct classes requiring different optimization strategies

Selenocysteine Insertion Sequence Binding Protein 2L Is Implicated as a Novel Post-Transcriptional Regulator of Selenoprotein Expression

The amino acid selenocysteine (Sec) is encoded by UGA codons. Recoding of UGA from stop to Sec requires a Sec insertion sequence (SECIS) element in the 3′ UTR of selenoprotein mRNAs. SECIS binding protein 2 (SBP2) binds the SECIS element and is essential for Sec incorporation into the nascent peptide. SBP2-like (SBP2L) is a paralogue of SBP2 in vertebrates and is the only SECIS binding protein in some invertebrates where it likely directs Sec incorporation. However, vertebrate SBP2L does not promote Sec incorporation in in vitro assays. Here we present a comparative analysis of SBP2 and SBP2L SECIS binding properties and demonstrate that its inability to promote Sec incorporation is not due to lower SECIS affinity but likely due to lack of a SECIS dependent domain association that is found in SBP2. Interestingly, however, we find that an invertebrate version of SBP2L is fully competent for Sec incorporation in vitro. Additionally, we present the first evidence that SBP2L interacts with selenoprotein mRNAs in mammalian cells, thereby implying a role in selenoprotein expression

Effect of Different Factors on Proliferation of Antler Cells, Cultured In Vitro

Antlers as a potential model for bone growth and development have become an object of rising interest. To elucidate processes explaining how antler growth is regulated, in vitro cultures have been established. However, until now, there has been no standard method to cultivate antler cells and in vitro results are often opposite to those reported in vivo. In addition, many factors which are often not taken into account under in vitro conditions may play an important role in the development of antler cells. In this study we investigated the effects of the antler growth stage, the male individuality, passaged versus primary cultures and the effect of foetal calf serum concentrations on proliferative potential of mixed antler cell cultures in vitro, derived from regenerating antlers of red deer males (Cervus elaphus). The proliferation potential of antler cells was measured by incorporation of 3H thymidine. Our results demonstrate that there is no significant effect of the antler growth stage, whereas male individuality and all other examined factors significantly affected antler cell proliferation. Furthermore, our results suggest that primary cultures may better represent in vivo conditions and processes occurring in regenerating antlers. In conclusion, before all main factors affecting antler cell proliferation in vitro will be satisfactorily investigated, results of in vitro studies focused on hormonal regulation of antler growth should be taken with extreme caution