Optimal Tree Research Articles

Fast Sparse Decision Tree Optimization via Reference Ensembles.

Sparse decision tree optimization has been one of the most fundamental problems in AI since its inception and is a challenge at the core of interpretable machine learning. Sparse decision tree optimization is computationally hard, and despite steady effort since the 1960's, breakthroughs have been made on the problem only within the past few years, primarily on the problem of finding optimal sparse decision trees. However, current state-of-the-art algorithms often require impractical amounts of computation time and memory to find optimal or near-optimal trees for some real-world datasets, particularly those having several continuous-valued features. Given that the search spaces of these decision tree optimization problems are massive, can we practically hope to find a sparse decision tree that competes in accuracy with a black box machine learning model? We address this problem via smart guessing strategies that can be applied to any optimal branch-and-bound-based decision tree algorithm. The guesses come from knowledge gleaned from black box models. We show that by using these guesses, we can reduce the run time by multiple orders of magnitude while providing bounds on how far the resulting trees can deviate from the black box's accuracy and expressive power. Our approach enables guesses about how to bin continuous features, the size of the tree, and lower bounds on the error for the optimal decision tree. Our experiments show that in many cases we can rapidly construct sparse decision trees that match the accuracy of black box models. To summarize: when you are having trouble optimizing, just guess.

Robust Optimal Classification Trees against Adversarial Examples

Decision trees are a popular choice of explainable model, but just like neural networks, they suffer from adversarial examples. Existing algorithms for fitting decision trees robust against adversarial examples are greedy heuristics and lack approximation guarantees. In this paper we propose ROCT, a collection of methods to train decision trees that are optimally robust against user-specified attack models. We show that the min-max optimization problem that arises in adversarial learning can be solved using a single minimization formulation for decision trees with 0-1 loss. We propose such formulations in Mixed-Integer Linear Programming and Maximum Satisfiability, which widely available solvers can optimize. We also present a method that determines the upper bound on adversarial accuracy for any model using bipartite matching. Our experimental results demonstrate that the existing heuristics achieve close to optimal scores while ROCT achieves state-of-the-art scores.

Probabilistic shaping PAM8 based on Huffman coding distribution matching in optical interconnection

We propose and experimentally verify a Huffman coding distribution matching (HCDM) scheme in intensity modulation and direct-detection (IM/DD) optical interconnection applying low-density parity-check-coded (LDPC) probabilistic shaping 8-level pulse amplitude modulation (PS-PAM8). Not only superior to that of conventional PS methods in view of arithmetic distribution matching (ADM) as well as constant composition distribution matching (CCDM) with much lower computational complexity, the HCDM can also enable arbitrary symbol probability distribution (SPD) by utilizing an optimal Huffman binary tree. The binomial coefficients required for SPD can be precomputed and stored in a lookup table. The experimental results show that the HCMD PS-PAM8 shows remarkably promoted receiver power sensitivity and improved system tolerance to optical fibre nonlinear effects compared to the uniformly distributed signal with the same effective net rate.

World-class interpretable poker

We address the problem of interpretability in iterative game solving for imperfect-information games such as poker. This lack of interpretability has two main sources: first, the use of an uninterpretable feature representation, and second, the use of black box methods such as neural networks, for the fitting procedure. In this paper, we present advances on both fronts. Namely, first we propose a novel, compact, and easy-to-understand game-state feature representation for Heads-up No-limit (HUNL) Poker. Second, we make use of globally optimal decision trees, paired with a counterfactual regret minimization (CFR) self-play algorithm, to train our poker bot which produces an entirely interpretable agent. Through experiments against Slumbot, the winner of the most recent Annual Computer Poker Competition, we demonstrate that our approach yields a HUNL Poker agent that is capable of beating the Slumbot. Most exciting of all, the resulting poker bot is highly interpretable, allowing humans to learn from the novel strategies it discovers.

Rigorous mathematical optimization of synthetic hepatic vascular trees.

In this paper, we introduce a new framework for generating synthetic vascular trees, based on rigorous model-based mathematical optimization. Our main contribution is the reformulation of finding the optimal global tree geometry into a nonlinear optimization problem (NLP). This rigorous mathematical formulation accommodates efficient solution algorithms such as the interior point method and allows us to easily change boundary conditions and constraints applied to the tree. Moreover, it creates trifurcations in addition to bifurcations. A second contribution is the addition of an optimization stage for the tree topology. Here, we combine constrained constructive optimization (CCO) with a heuristic approach to search among possible tree topologies. We combine the NLP formulation and the topology optimization into a single algorithmic approach. Finally, we attempt the validation of our new model-based optimization framework using a detailed corrosion cast of a human liver, which allows a quantitative comparison of the synthetic tree structure with the tree structure determined experimentally down to the fifth generation. The results show that our new framework is capable of generating asymmetric synthetic trees that match the available physiological corrosion cast data better than trees generated by the standard CCO approach.

Using Artificial Intelligence to Find the Optimal Margin Width in Hepatectomy for Colorectal Cancer Liver Metastases

In patients with resectable colorectal cancer liver metastases (CRLM), the choice of surgical technique and resection margin are the only variables that are under the surgeon's direct control and may influence oncologic outcomes. There is currently no consensus on the optimal margin width. To determine the optimal margin width in CRLM by using artificial intelligence-based techniques developed by the Massachusetts Institute of Technology and to assess whether optimal margin width should be individualized based on patient characteristics. The internal cohort of the study included patients who underwent curative-intent surgery for KRAS-variant CRLM between January 1, 2000, and December 31, 2017, at Johns Hopkins Hospital, Baltimore, Maryland, Memorial Sloan Kettering Cancer Center, New York, New York, and Charité-University of Berlin, Berlin, Germany. Patients from institutions in France, Norway, the US, Austria, Argentina, and Japan were retrospectively identified from institutional databases and formed the external cohort of the study. Data were analyzed from April 15, 2019, to November 11, 2021. Hepatectomy. Patients with KRAS-variant CRLM who underwent surgery between 2000 and 2017 at 3 tertiary centers formed the internal cohort (training and testing). In the training cohort, an artificial intelligence-based technique called optimal policy trees (OPTs) was used by building on random forest (RF) predictive models to infer the margin width associated with the maximal decrease in death probability for a given patient (ie, optimal margin width). The RF component was validated by calculating its area under the curve (AUC) in the testing cohort, whereas the OPT component was validated by a game theory-based approach called Shapley additive explanations (SHAP). Patients from international institutions formed an external validation cohort, and a new RF model was trained to externally validate the OPT-based optimal margin values. This cohort study included a total of 1843 patients (internal cohort, 965; external cohort, 878). The internal cohort included 386 patients (median [IQR] age, 58.3 [49.0-68.7] years; 200 men [51.8%]) with KRAS-variant tumors. The AUC of the RF counterfactual model was 0.76 in both the internal training and testing cohorts, which is the highest ever reported. The recommended optimal margin widths for patient subgroups A, B, C, and D were 6, 7, 12, and 7 mm, respectively. The SHAP analysis largely confirmed this by suggesting 6 to 7 mm for subgroup A, 7 mm for subgroup B, 7 to 8 mm for subgroup C, and 7 mm for subgroup D. The external cohort included 375 patients (median [IQR] age, 61.0 [53.0-70.0] years; 218 men [58.1%]) with KRAS-variant tumors. The new RF model had an AUC of 0.78, which allowed for a reliable external validation of the OPT-based optimal margin. The external validation was successful as it confirmed the association of the optimal margin width of 7 mm with a considerable prolongation of survival in the external cohort. This cohort study used artificial intelligence-based methodologies to provide a possible resolution to the long-standing debate on optimal margin width in CRLM.

Toddler Nutritional Status Classification Using C4.5 and Particle Swarm Optimization

Abstract. Purpose: This research was conducted to create a classification model in the form of the most optimal decision tree. Optimal in this case is the combination of parameters used that will produce the highest accuracy compared to other parameter combinations. From this best model, it will be used to predict the nutritional status class for the new data.Methods/Study design/approach: The dataset used is from Nutritional Status Monitoring in 2017 in Riau Province, Indonesia. From the dataset, the Knowledge Discovery in Database (KDD) stages were carried out to build several classification models in the form of decision trees. The decision tree that has the highest accuracy will then be selected to predict the class for the new data. Predictions for new data (unclassified data) will be made in a web-based system.Result/Findings: Particle Swarm Optimization is used to find optimal parameters. Before PSO is used, there are 213 parameters in the dataset that can be used to do classification. However, using many such parameters is time-consuming. After PSO is used, the optimal parameters found are the combination of 4 parameters, which can produce the most optimal decision tree. The 4 chosen parameters are gender, age (in months), height, and the way to measure the height (either stand up or lie down). The most optimal decision tree has an accuracy of 94.49%. From the most optimal decision tree, a web-based system was built to predict the class for new data (unclassified data).Novelty/Originality/Value: Particle Swarm Optimization (PSO) is a method that can help to select the most optimal parameters, or in other words produce the highest classification accuracy. The combination of parameters selected has also been confirmed by the nutritionist. The prediction system has been declared feasible to be used by nutritionists through the User Acceptance Test (UAT).

A New Method for Reconstructing Data Considering the Factor of Selected Provider Nodes Set in Distributed Storage System

In the distributed storage system, when data need to be recovered after node failure, the erasure code redundancy method occupies less storage space than the multi-copy method. At present, the repair mechanism using erasure code to reconstruct the failed node only considers the improvement of link bandwidth on the repair rate and does not consider the impact of the selection of data providing node-set on the repair performance. A single node fault data reconstruction method based on the Software Defined Network (SDN) using the erasure code method is designed to solve the above problems. This method collects the network link-state through SDN, establishes a multi-attribute decision-making model of the data providing node-set based on the node performance, and determines the data providing nodes participating in providing data through the ideal point method. Then, the data recovery problem of a single fault node is modeled as the optimization problem of an optimal repair tree, and a hybrid genetic algorithm is designed to solve it. The experimental results show that under the same erasure code scale, after selecting the nodes of the data providing node-set, compared with the traditional tree topology and star topology, the repair delay distribution of the designed single fault node repair method for a distributed storage system is reduced by 15% and 45% respectively, and the repair flow is close to the star topology, which is reduced by 40% compared with the traditional tree repair.

Semantic segmentation of building extraction in very high resolution imagery via optimal segmentation guided by deep seeds

The automatic segmentation of buildings from satellite images is extremely challenging due to the complex shapes of buildings. Deep convolutional neural networks (DCNNs) have recently enabled accurate pixel-level labeling tasks, which ensures precise boundaries. However, the large number of pixel-level labels required for DCNN-based semantic segmentation of buildings represents a considerable obstacle to the process. We propose a framework that relies on deep seeds and optimal segmentation to extract buildings from very high resolution imagery to solve the aforementioned issue. Input images are cropped rather than resized to a low-resolution image and passed directly to DCNN for segmentation. An image resize transformation will lead to a significant loss of image detail, especially for images with high resolution. For both image patches and resized images, a classification network is utilized to locate deep seeds within building and nonbuilding classes. Then, a boundary map can be predicted by passing the resized image through a convolutional oriented boundary network. A hierarchical segmentation tree is built to determine the optimal segmentation by seeking the optimal tree cut. The final segmentation is achieved by developing a graphic model representing a set of nodes that distribute information from deep seeds to unmarked regions. According to the experiments conducted on the ISPRS two-dimensional semantic labeling contest (Potsdam) and the WHU building datasets, the proposed framework is able to significantly enhance the building segmentation accuracy. We have achieved such improvements through a cost-effective computing method without training a segmentation network.

Embedding gene trees into phylogenetic networks by conflict resolution algorithms

BackgroundPhylogenetic networks are mathematical models of evolutionary processes involving reticulate events such as hybridization, recombination, or horizontal gene transfer. One of the crucial notions in phylogenetic network modelling is displayed tree, which is obtained from a network by removing a set of reticulation edges. Displayed trees may represent an evolutionary history of a gene family if the evolution is shaped by reticulation events.ResultsWe address the problem of inferring an optimal tree displayed by a network, given a gene tree G and a tree-child network N, under the deep coalescence and duplication costs. We propose an O(mn)-time dynamic programming algorithm (DP) to compute a lower bound of the optimal displayed tree cost, where m and n are the sizes of G and N, respectively. In addition, our algorithm can verify whether the solution is exact. Moreover, it provides a set of reticulation edges corresponding to the obtained cost. If the cost is exact, the set induces an optimal displayed tree. Otherwise, the set contains pairs of conflicting edges, i.e., edges sharing a reticulation node. Next, we show a conflict resolution algorithm that requires 2^{r+1}-1 invocations of DP in the worst case, where r is the number of reticulations. We propose a similar O(2^kmn)-time algorithm for level-k tree-child networks and a branch and bound solution to compute lower and upper bounds of optimal costs. We also extend the algorithms to a broader class of phylogenetic networks. Based on simulated data, the average runtime is Theta (2^{{0.543}k}mn) under the deep-coalescence cost and Theta (2^{{0.355}k}mn) under the duplication cost.ConclusionsDespite exponential complexity in the worst case, our algorithms perform significantly well on empirical and simulated datasets, due to the strategy of resolving internal dissimilarities between gene trees and networks. Therefore, the algorithms are efficient alternatives to enumeration strategies commonly proposed in the literature and enable analyses of complex networks with dozens of reticulations.

Accuracy of Risk Estimation for Surgeons Versus Risk Calculators in Emergency General Surgery

IntroductionSurgical risk calculators have expanded in both number and sophistication of their predictive approach. These calculators are gaining popularity as validated tools to help surgeons estimate mortality and complications following emergency general surgery (EGS). However, the accuracy of risk estimates generated by these calculators compared to risk estimation by practicing surgeons has not been explored. MethodsAcute care surgeons at a quaternary care center prospectively estimated 30-d mortality and complications for adult EGS patients (2019-2021). Surgeon predictions were compared to Predictive OpTimal Trees in Emergency Surgery Risk (POTTER) and NSQIP estimates. Observed-to-expected (O:E) ratios of median aggregate estimates were calculated. C-statistics for surgeon and calculator estimations were utilized to quantify predictive accuracy. ResultsAmong 150 patients (median 61 y, 45% male), 30-d mortality was 15% (n = 23). Observed rates of prolonged mechanical ventilation and acute renal failures were 30% and 10%, respectively. Overall, surgeon predictions were similar to risk calculator estimates for mortality (c-statistics 0.843 [surgeon] versus 0.848 [POTTER] and 0.815 [NSQIP]) and need for prolonged ventilation (c-statistics 0.801 versus 0.722 and 0.689, respectively). Surgeons tended to overestimate complication risks. Surgeon experience was not significantly associated with mortality prediction in an adjusted model. ConclusionsAcute care surgeons at a quaternary care center predicted postoperative mortality and complications with similar discrimination when compared to surgical risk calculators. Surgeon expertise should be utilized in conjunction with risk calculators when counseling EGS patients regarding anticipated postoperative outcomes. Surgeons should be cognizant of patterns in overestimation or underestimation of complications.

Power flow management of hybrid system in smart grid requirements using ITSA-MOAT approach

A power flow management (PFM) concept of Photovoltaic/Fuel Cell/Battery/Super capacitor in smart grid (SG) system is proposed in this paper. The proposed system controls the photovoltaic (PV) system, battery storage, fuel cell (FC) and super capacitor (SC). The proposed control system is the combination of Improved Tunicate Swarm Optimization (ITSA) and Multi Objective Artificial Tree (MOAT), hence it is called ITSA-MOAT method. Here, ITSA is building up the control pulses of the inverter using the energy exchange assortment between the source and load side. Here, the composition of multi-objective function is considered according to the obtainable source power and several specified grid produced by the active power and reactive power. To acquire the online control pulses, MOAT is utilized based on the variants of power. Moreover, the global state of charge (SoC) of energy storages and load demand are considered. The ITSA-MOAT method is implemented in MATLAB/Simulink platform. By then, the experimental results of the proposed method are analyzed with the existing methods.

Using Machine Learning to Identify the Optimal Limb Symmetry Index Cut-Off Threshold in Paediatric Patients with ACL Injury

Background:The risk for a subsequent ACL injury following a primary paediatric ACL reconstruction is extremely high, with 17-30% of paediatric patients sustaining a second ACL injury within two years. It is possible that this risk is associated with a lack of evidence for identifying return-to-activity (RTA) measures to guide postoperative decision-making. Limb symmetry indices (LSI) derived from functional and strength tasks are commonly used, however LSIs used by paediatric orthopaedic surgeons can range from 75-95%, with no objective evidence to support these values.Purpose:To create knowledge that could be used to inform RTA and to determine if a machine learning approach could objectively identify LSI thresholds from functional tasks that classify paediatric individuals as injured or healthy.Methods:Forty-two patients (30 females) who had suffered an ACL injury (ACLi) and 69 matched uninjured controls (36 females; CON) performed isometric knee extension and flexion, and single-leg hopping tasks. LSIs were calculated as injured/uninjured limb for ACLi and non-dominant/dominant limb for CON. LSIs with significant between-group (ACLi vs CON) differences (independent t-tests), were used in a machine learning algorithm. A classification tree (CT) was used to classify ACLi and CON participants using LSI percentages. Ten-fold cross-validation was used to determine optimal tree complexity. LSIs were then converted from percentage to categorical ‘pass/fail’ grades according to cut-off thresholds ranging from 70-100%. CTs were then fit to the converted categorical data at each cut-off threshold and the model accuracies were compared.Results:Anterior, lateral, triple, and timed 6m hop tasks, and knee extension strength LSIs were used in the machine learning algorithm. The CT using the raw LSI percentages correctly classified 85.9% of participants. When data were converted from percentage to categorical ‘pass/fail’ grades, a plateau in classification accuracy occurred between LSIs cut-off thresholds of ˜83-90%. The most accurate LSI cut-off threshold was 89%, which correctly classified 84.4% of participants.Conclusion:Based on our preliminary analysis, the 90% LSI cut-off is an appropriate threshold for separating the hopping and strength performance of ACLi and CON paediatric participants. It is nevertheless important to recognize that this threshold can identify those with ACL injury, and not those who are ready for return to activity. A 90% LSI is therefore only a minimum criterion to reach as part of an RTA evaluation, and not a threshold that identifies when an individual is ready to return.

Restricted Hamming–Huffman trees

We study a special case of Hamming–Huffman trees, in which both data compression and data error detection are tackled on the same structure. Given a hypercubeQnof dimensionn, we are interested in some aspects of its vertex neighborhoods. For a subsetLof vertices ofQn, the neighborhood ofLis defined as the union of the neighborhoods of the vertices ofL. The minimum neighborhood problem is that of determining the minimum neighborhood cardinality over all those setsL. This is a well-known problem that has already been solved. Our interest lies in determining optimal Hamming–Huffman trees, a problem that remains open and which is related to minimum neighborhoods inQn. In this work, we consider a restricted version of Hamming–Huffman trees, called [k]-HHT s, which admit symbol leaves in at mostkdifferent levels. We present an algorithm to build optimal [2]-HHT s. For uniform frequencies, we prove that an optimal HHT is always a [5]-HHT and that there exists an optimal HHT which is a [4]-HHT . Also, considering experimental results, we conjecture that there exists an optimal tree which is a [3]-HHT .

POTTER-ICU: An artificial intelligence smartphone-accessible tool to predict the need for intensive care after emergency surgery

BackgroundDelays in admitting high-risk emergency surgery patients to the intensive care unit result in worse outcomes and increased health care costs. We aimed to use interpretable artificial intelligence technology to create a preoperative predictor for postoperative intensive care unit need in emergency surgery patients. MethodsA novel, interpretable artificial intelligence technology called optimal classification trees was leveraged in an 80:20 train:test split of adult emergency surgery patients in the 2007–2017 American College of Surgeons National Surgical Quality Improvement Program database. Demographics, comorbidities, and laboratory values were used to develop, train, and then validate optimal classification tree algorithms to predict the need for postoperative intensive care unit admission. The latter was defined as postoperative death or the development of 1 or more postoperative complications warranting critical care (eg, unplanned intubation, ventilator requirement ≥48 hours, cardiac arrest requiring cardiopulmonary resuscitation, and septic shock). An interactive and user-friendly application was created. C statistics were used to measure performance. ResultsA total of 464,861 patients were included. The mean age was 55 years, 48% were male, and 11% developed severe postoperative complications warranting critical care. The Predictive OpTimal Trees in Emergency Surgery Risk Intensive Care Unit application was created as the user-friendly interface of the complex optimal classification tree algorithms. The number of questions (ie, tree depths) needed to predict intensive care unit admission ranged from 2 to 11. The Predictive OpTimal Trees in Emergency Surgery Risk Intensive Care Unit application had excellent discrimination for predicting the need for intensive care unit admission (C statistics: 0.89 train, 0.88 test). ConclusionWe recommend the Predictive OpTimal Trees in Emergency Surgery Risk Intensive Care Unit application as an accurate, artificial intelligence–based tool for predicting severe complications warranting intensive care unit admission after emergency surgery. The Predictive OpTimal Trees in Emergency Surgery Risk Intensive Care Unit application can prove useful to triage patients to the intensive care unit and to potentially decrease failure to rescue in emergency surgery patients.

Recognition of Immune Cell Markers of COVID-19 Severity with Machine Learning Methods.

COVID-19 is hypothesized to be linked to the host's excessive inflammatory immunological response to SARS-CoV-2 infection, which is regarded to be a major factor in disease severity and mortality. Numerous immune cells play a key role in immune response regulation, and gene expression analysis in these cells could be a useful method for studying disease states, assessing immunological responses, and detecting biomarkers. Here, we developed a machine learning procedure to find biomarkers that discriminate disease severity in individual immune cells (B cell, CD4+ cell, CD8+ cell, monocyte, and NK cell) using single-cell gene expression profiles of COVID-19. The gene features of each profile were first filtered and ranked using the Boruta feature selection method and mRMR, and the resulting ranked feature lists were then fed into the incremental feature selection method to determine the optimal number of features with decision tree and random forest algorithms. Meanwhile, we extracted the classification rules in each cell type from the optimal decision tree classifiers. The best gene sets discovered in this study were analyzed by GO and KEGG pathway enrichment, and some important biomarkers like TLR2, ITK, CX3CR1, IL1B, and PRDM1 were validated by recent literature. The findings reveal that the optimal gene sets for each cell type can accurately classify COVID-19 disease severity and provide insight into the molecular mechanisms involved in disease progression.

Spatial connectivity in tree-level decision-support models using mathematical optimization and individual tree mapping

Tree-level planning is gaining relevance supported by continuous advances in forest remote sensing methods. Spatial connectivity under spatial optimization methods applied to individual tree mapping data remains unexplored. This article presents a spatially explicit mathematical formulation that ensures the tree cuttings conforming a corridor can connect two points across a forest landscape. The created paths facilitate operational harvesting linking forest planning solutions to forest machinery operations. The model integrates production, economic and connectivity constraints using mixed integer programming as optimization method, while tree positions and attributes were mapped using airborne laser scanning. Model performance was tested towards different harvesting targets and for the assessment of trade-offs involving the connectivity of the solutions and the economic performance. The showcase area to test the model is pine forest located in Central Spain comprising more than 9000 detected trees. The results showed the mathematical formulation in the model is effective at creating paths connect proposed areas using optimized tree harvests. Global optimality was reached fast using mathematical programming for a complex and large combinatorial problem. The presented model is another step forward in the design of multi-objective tree-level planning models, enhancing the assimilation of planning solutions towards forest operations capable to maximize the use of individual tree mapping data.

Optimal survival trees

Tree-based models are increasingly popular due to their ability to identify complex relationships that are beyond the scope of parametric models. Survival tree methods adapt these models to allow for the analysis of censored outcomes, which often appear in medical data. We present a new Optimal Survival Trees algorithm that leverages mixed-integer optimization (MIO) and local search techniques to generate globally optimized survival tree models. We demonstrate that the OST algorithm improves on the accuracy of existing survival tree methods, particularly in large datasets.

Study on Model of Natural Gas Hydrate Formation Based on Extremely Randomized Trees

Abstract It is easy to form hydrates in the development of natural gas. It is of great significance to study the formation of prediction models to guide the safe production of oil and gas fields. Based on computer intelligence algorithms, a prediction model of natural gas hydrate formation based on Extremely randomized trees was established and compared with the BP Neral Network model. To objectively evaluate the predictive power of the model, an extensive database of more than 1000 hydrate formation conditions was established. The results show that the number of optimal decision trees for the Extremely randomized trees model is 6, and the decision tree depth is 32. The BP Neral Network model has a flat error distribution with a maximum error of 6.37%. The error distribution of the Extremely randomized trees model is abrupt, with a maximum error of 3.39%, with higher stability and accuracy. In terms of pure water, the BP Neral Network model performs well only in a small number of conditions due to over-fitting, but the Extremely randomized trees model can avoid over-fitting by using the large number theorem, showing a stronger advantage.

Decorated merge trees for persistent topology

This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine both $$\pi _0$$ and $$H_n$$ information into a single data structure that distinguishes filtrations that merge trees and persistent homology cannot distinguish alone. Three variants on DMTs, which emphasize category theory, representation theory and persistence barcodes, respectively, offer different advantages in terms of theory and computation. Two notions of distance—an interleaving distance and bottleneck distance—for DMTs are defined and a hierarchy of stability results that both refine and generalize existing stability results is proved here. To overcome some of the computational complexity inherent in these distances, we provide a novel use of Gromov-Wasserstein couplings to compute optimal merge tree alignments for a combinatorial version of our interleaving distance which can be tractably estimated. We introduce computational frameworks for generating, visualizing and comparing decorated merge trees derived from synthetic and real data. Example applications include comparison of point clouds, interpretation of persistent homology of sliding window embeddings of time series, visualization of topological features in segmented brain tumor images and topology-driven graph alignment.