Topological Data Analysis in Investment Decisions Abstract   This article explores the applications of Topological Data Analysis (TDA) in the finance field, especially addressing the primordial problem of asset allocation. Firstly, we build a rationale on why TDA can be a better alternative to traditional risk indicators such as standard deviation using real data sets. We apply Takens embedding theorem to reconstruct the time series of returns in a high dimensional space. We adopt the sliding window approach to draw the timedependent point cloud data sets and associate a topological space with them. We then apply the persistent homology to discover the topological patterns that appear in the multidimensional time series. The temporal changes in the persistence landscapes, which are the realvalued functions that encode the persistence of topological patterns, are captured via Lp norm. The time series of the Lp norms shows that it is better at measuring the dynamics of returns than the standard deviation.  Inspired by our findings, we explore an application of TDA in Enhanced Indexing (EI) that aims to build a portfolio of fewer assets than that in the index to outperform the latter. We propose a twostep procedure to accomplish this task. In step one, we utilize the Lp norms of the assets to propose a filtration technique of selecting a few assets from a larger pool of assets. In step two, we propose an optimization model to construct an optimal portfolio from the class of filtered assets for EI. To test the efficiency of this enhanced algorithm, experiments are carried out on ten data sets from financial markets across the globe. Our extensive empirical analysis exhibits that the proposed strategy delivers superior performance on several measures, including excess mean returns from the benchmark index and tail rewardrisk ratios than some of the existing models of EI in the literature. The proposed filtering strategy is also noted to be beneficial for both riskseeking and riskaverse investors.

Robust Graph Topology Learning and Application in Stock Market Inference In many applications, there are multiple interacting entities, generating time series of data over the space. To describe the relation within the set of data, the underlying topology may be used. In many real applications, not only the signaldata of interest is measured in noise, but it is also contaminated with outliers. The proposed method, called RGTL, infers the graph topology from noisy measurements and removes these outliers simultaneously. Here, it is assumed that we have no information about the space graph topology, while we know that graph signal are sampled consecutively in time and thus the graph in time domain is given. The simulation results show that the proposed algorithm has a better performance for different graph orders, compared with the conventional graph topology inference methods. Due to the nature of stock market data and the presence of noise and high power outliers, the proposed method is also applied to find some relations among selected ticker symbol prices in the USA market.