arXiv subjects

Woojin Kim

Publications and source records attributed to Woojin Kim.

33 records · Page 2Linked to original sources

arXiv2024

The Generalized Rank Invariant: Möbius invertibility, Discriminating Power, and Connection to Other Invariants

In addition to inherent computational challenges, the absence of a canonical method for quantifying `persistence' in multi-parameter persistent homology remains a hurdle in its application. One of the best known quantifications of persistence for multi-parameter persistent homology is the rank invariant, which has recently evolved into the generalized rank invariant (GRI) by naturally extending its domain. This extension enables us to quantify persistence across a broader range of regions in the indexing poset compared to the rank invariant. However, the size of the domain of the GRI is generally formidable, making it desirable to restrict its domain to a more manageable subset for computational purposes. The foremost questions regarding such a restriction of the domain are: (1) How to restrict, if possible, the domain of the GRI without any loss of information? (2) When can we more compactly encode the GRI as a `persistence diagram'? (3) What is the trade-off between computational efficiency and the discriminating power of the GRI as the amount of the restriction on the domain varies? (4) What proxies exist for persistence diagrams in the multi-parameter setting that can be derived from the GRI? To address the first three questions, we generalize and axiomatize the classic fundamental lemma of persistent homology via the notion of Möbius invertibility of the GRI which we propose. This extension also contextualizes known results regarding the (generalized) rank invariant within the classical theory of Möbius inversion. We conduct a comprehensive comparison between Möbius invertibility and other existing concepts related to the structural simplicity of persistence modules. We address the fourth question through the notion of motivic invariants. We demonstrate that many invariants from the literature can be both derived from the GRI and recast as motivic invariants.

math.AT↗

arXiv2024

Bigraded Betti numbers and Generalized Persistence Diagrams

Commutative diagrams of vector spaces and linear maps over $\mathbb{Z}^2$ are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation theory tells us that such diagrams are of wild type, studying informative invariants of a 2-parameter persistence module $M$ is of central importance in TDA. One of such invariants is the generalized rank invariant, recently introduced by Kim and Mémoli. Via the Möbius inversion of the generalized rank invariant of $M$, we obtain a collection of connected subsets $I\subset\mathbb{Z}^2$ with signed multiplicities. This collection generalizes the well known notion of persistence barcode of a persistence module over $\mathbb{R}$ from TDA. In this paper we show that the bigraded Betti numbers of $M$, a classical algebraic invariant of $M$, are obtained by counting the corner points of these subsets $I$s. Along the way, we verify that an invariant of 2-parameter persistence modules called the interval decomposable approximation (introduced by Asashiba et al.) also encodes the bigraded Betti numbers in a similar fashion. We also show that the aforementioned results are optimal in the sense that they cannot be extended to $d$-parameter persistence modules for $d \geq 3$.

math.AT↗

arXiv2023

MKConv: Multidimensional Feature Representation for Point Cloud Analysis

Despite the remarkable success of deep learning, an optimal convolution operation on point clouds remains elusive owing to their irregular data structure. Existing methods mainly focus on designing an effective continuous kernel function that can handle an arbitrary point in continuous space. Various approaches exhibiting high performance have been proposed, but we observe that the standard pointwise feature is represented by 1D channels and can become more informative when its representation involves additional spatial feature dimensions. In this paper, we present Multidimensional Kernel Convolution (MKConv), a novel convolution operator that learns to transform the point feature representation from a vector to a multidimensional matrix. Unlike standard point convolution, MKConv proceeds via two steps. (i) It first activates the spatial dimensions of local feature representation by exploiting multidimensional kernel weights. These spatially expanded features can represent their embedded information through spatial correlation as well as channel correlation in feature space, carrying more detailed local structure information. (ii) Then, discrete convolutions are applied to the multidimensional features which can be regarded as a grid-structured matrix. In this way, we can utilize the discrete convolutions for point cloud data without voxelization that suffers from information loss. Furthermore, we propose a spatial attention module, Multidimensional Local Attention (MLA), to provide comprehensive structure awareness within the local point set by reweighting the spatial feature dimensions. We demonstrate that MKConv has excellent applicability to point cloud processing tasks including object classification, object part segmentation, and scene semantic segmentation with superior results.

cs.CV↗

arXiv2022

Direct visualization and control of SrOx segregation on semiconducting Nb doped SrTiO3 (100) surface

We investigated how SrOx segregates on a Nb doped SrTiO3 (100) surface by in air annealing. Using atomic force and photoemission electron microscopes, we can directly visualize the morphology and the electronic phase changes with SrOx segregation. SrOx islands less than 2 micron meter in size and 1-5 unit cells thick nucleate first and grow in a labyrinth domain pattern. After prolonged annealing, SrOx forms a ~10 nm thick film. We show that the domain pattern can be controlled by introducing a surface miscut angle of SrTiO3. Additionally, the segregated SrOx has a lower work function, compared to that of SrTiO3. These results suggest that the control and tunability of SrOx segregation is applicable to the design of a new functional electronic devices in the semiconducting SrTiO3 based heterostructure.

cond-mat.mtrl-sci↗

arXiv2022

Interleaving by Parts: Join Decompositions of Interleavings and Join-Assemblage of Geodesics

Metrics of interest in topological data analysis (TDA) are often explicitly or implicitly in the form of an interleaving distance $d_{\mathrm{I}}$ between poset maps (i.e. order-preserving maps), e.g. the Gromov-Hausdorff distance between metric spaces can be reformulated in this way. We propose a representation of a poset map $\mathbf{F}:\mathcal{P}\to\mathcal{Q}$ as a join (i.e. supremum) $\bigvee_{b\in B} \mathbf{F}_b$ of simpler poset maps $\mathbf{F}_b$ (for a join dense subset $B\subset \mathcal{Q}$) which in turn yields a decomposition of $d_{\mathrm{I}}$ into a product metric. The decomposition of $d_{\mathrm{I}}$ is simple, but its ramifications are manifold: (1) We can construct a geodesic path between any poset maps $\mathbf{F}$ and $\mathbf{G}$ with $d_{\mathrm{I}}(\mathbf{F},\mathbf{G})<\infty$ by assembling geodesics between all $\mathbf{F}_b$s and $\mathbf{G}_b$s via the join operation. This construction generalizes at least three constructions of geodesic paths that have appeared in the literature. (2) We can extend the Gromov-Hausdorff distance to a distance between simplicial filtrations over an arbitrary poset with a flow, preserving its universality and geodesicity. (3) We can clarify equivalence between several known metrics on multiparameter hierarchical clusterings. (4) We can illuminate the relationship between the erosion distance by Patel and the graded rank function by Betthauser, Bubenik, and Edwards, which in turn takes us to an interpretation on the representation $\bigvee_b \mathbf{F}_b$ as a generalization of persistence landscapes and graded rank functions.

math.AT↗

arXiv2022

Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and its Applications

The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval $I$ of a $\mathbb{Z}^2$-indexed persistence module $M$ is equal to the generalized rank of the zigzag module that is induced on a certain path in $I$ tracing mostly its boundary. Hence, we can compute the generalized rank over $I$ by computing the barcode of the zigzag module obtained by restricting the bifiltration inducing $M$ to that path. If the bifiltration and $I$ have at most $t$ simplices and points respectively, this computation takes $O(t^ω)$ time where $ω\in[2,2.373)$ is the exponent of matrix multiplication. Among others, we apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module $M$, determine whether $M$ is interval decomposable and, if so, compute all intervals supporting its summands.

math.AT↗

arXiv2022

Extracting Persistent Clusters in Dynamic Data via Möbius inversion

Identifying and representing clusters in time-varying network data is of particular importance when studying collective behaviors emerging in nature, in mobile device networks or in social networks. Based on combinatorial, categorical, and persistence theoretic viewpoints, we establish a stable functorial pipeline for the summarization of the evolution of clusters in a time-varying network. We first construct a complete summary of the evolution of clusters in a given time-varying network over a set of entities $X$ of which takes the form of a formigram. This formigram can be understood as a certain Reeb graph $\mathcal{R}$ which is labeled by subsets of $X$. By applying Möbius inversion to the formigram in two different manners, we obtain two dual notions of diagram: the maximal group diagram and the persistence clustergram, both of which are in the form of an `annotated' barcode. The maximal group diagram consists of time intervals annotated by their corresponding maximal groups -- a notion due to Buchin et al., implying that we recognize the notion of maximal groups as a special instance of generalized persistence diagram by Patel. On the other hand, the persistence clustergram is mostly obtained by annotating the intervals in the zigzag barcode of the Reeb graph $\mathcal{R}$ with certain merging/disbanding events in the given time-varying network. We show that both diagrams are complete invariants of formigrams (or equivalently of trajectory grouping structure by Buchin et al.) and thus contain more information than the Reeb graph $\mathcal{R}$.

math.AT↗

arXiv2021

Robust Lane Detection via Expanded Self Attention

The image-based lane detection algorithm is one of the key technologies in autonomous vehicles. Modern deep learning methods achieve high performance in lane detection, but it is still difficult to accurately detect lanes in challenging situations such as congested roads and extreme lighting conditions. To be robust on these challenging situations, it is important to extract global contextual information even from limited visual cues. In this paper, we propose a simple but powerful self-attention mechanism optimized for lane detection called the Expanded Self Attention (ESA) module. Inspired by the simple geometric structure of lanes, the proposed method predicts the confidence of a lane along the vertical and horizontal directions in an image. The prediction of the confidence enables estimating occluded locations by extracting global contextual information. ESA module can be easily implemented and applied to any encoder-decoder-based model without increasing the inference time. The performance of our method is evaluated on three popular lane detection benchmarks (TuSimple, CULane and BDD100K). We achieve state-of-the-art performance in CULane and BDD100K and distinct improvement on TuSimple dataset. The experimental results show that our approach is robust to occlusion and extreme lighting conditions.

cs.CV↗

arXiv2021

Generalized Persistence Diagrams for Persistence Modules over Posets

When a category $\mathcal{C}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P} \rightarrow \mathcal{C}$ from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules $F:\mathbf{P} \rightarrow \mathbf{vec}$ of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset $\mathbf{P}$ of $F: \mathbf{P} \rightarrow \mathcal{C}$ in defining Patel's generalized persistence diagram of $F$. Of particular importance is the fact that the generalized persistence diagram of $F$ is defined regardless of whether $F$ is interval decomposable or not. By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type $\mathcal{A}$ persistence diagram to Lipschitz continuity theorem for the category of sets.

math.AT↗

arXiv2021

Driver Behavior Recognition via Interwoven Deep Convolutional Neural Nets with Multi-stream Inputs

Understanding driver activity is vital for in-vehicle systems that aim to reduce the incidence of car accidents rooted in cognitive distraction. Automating real-time behavior recognition while ensuring actions classification with high accuracy is however challenging, given the multitude of circumstances surrounding drivers, the unique traits of individuals, and the computational constraints imposed by in-vehicle embedded platforms. Prior work fails to jointly meet these runtime/accuracy requirements and mostly rely on a single sensing modality, which in turn can be a single point of failure. In this paper, we harness the exceptional feature extraction abilities of deep learning and propose a dedicated Interwoven Deep Convolutional Neural Network (InterCNN) architecture to tackle the problem of accurate classification of driver behaviors in real-time. The proposed solution exploits information from multi-stream inputs, i.e., in-vehicle cameras with different fields of view and optical flows computed based on recorded images, and merges through multiple fusion layers abstract features that it extracts. This builds a tight ensembling system, which significantly improves the robustness of the model. In addition, we introduce a temporal voting scheme based on historical inference instances, to enhance the classification accuracy. Experiments conducted with a dataset that we collect in a mock-up car environment demonstrate that the proposed InterCNN with MobileNet convolutional blocks can classify 9 different behaviors with 73.97% accuracy, and 5 'aggregated' behaviors with 81.66% accuracy. We further show that our architecture is highly computationally efficient, as it performs inferences within 15ms, which satisfies the real-time constraints of intelligent cars. Nevertheless, our InterCNN is robust to lossy input, as the classification remains accurate when two input streams are occluded.

cs.CV↗

arXiv2020

Elder-Rule-Staircodes for Augmented Metric Spaces

An augmented metric space is a metric space $(X, d_X)$ equipped with a function $f_X: X \to \mathbb{R}$. This type of data arises commonly in practice, e.g, a point cloud $X$ in $\mathbb{R}^d$ where each point $x\in X$ has a density function value $f_X(x)$ associated to it. An augmented metric space $(X, d_X, f_X)$ naturally gives rise to a 2-parameter filtration $\mathcal{K}$. However, the resulting 2-parameter persistent homology $\mathrm{H}_{\bullet}(\mathcal{K})$ could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode $\mathrm{H}_0(\mathcal{K})$. Specifically, if $n = |X|$, the elder-rule-staircode consists of $n$ number of staircase-like blocks in the plane. We show that if $\mathrm{H}_0(\mathcal{K})$ is interval decomposable, then the barcode of $\mathrm{H}_0(\mathcal{K})$ is equal to the elder-rule-staircode. Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), and the graded Betti numbers of $\mathrm{H}_0(\mathcal{K})$ can all be efficiently computed once the elder-rule-staircode is given. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in $O(n^2\log n)$ time, which can be improved to $O(n^2α(n))$ if $X$ is from a fixed dimensional Euclidean space $\mathbb{R}^d$, where $α(n)$ is the inverse Ackermann function.

math.AT↗

arXiv2020

False Positive Removal for 3D Vehicle Detection with Penetrated Point Classifier

Recently, researchers have been leveraging LiDAR point cloud for higher accuracy in 3D vehicle detection. Most state-of-the-art methods are deep learning based, but are easily affected by the number of points generated on the object. This vulnerability leads to numerous false positive boxes at high recall positions, where objects are occasionally predicted with few points. To address the issue, we introduce Penetrated Point Classifier (PPC) based on the underlying property of LiDAR that points cannot be generated behind vehicles. It determines whether a point exists behind the vehicle of the predicted box, and if does, the box is distinguished as false positive. Our straightforward yet unprecedented approach is evaluated on KITTI dataset and achieved performance improvement of PointRCNN, one of the state-of-the-art methods. The experiment results show that precision at the highest recall position is dramatically increased by 15.46 percentage points and 14.63 percentage points on the moderate and hard difficulty of car class, respectively.

cs.CV↗

arXiv2019

Spatio-temporal Persistent Homology for Dynamic Metric Spaces

Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter "spatiotemporal" filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov-Hausdorff distance which permits metrizing the collection of all DMSs. On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well known interleaving distance. We also study the computational complexity associated to both choices of d.

math.AT↗

arXiv2008

Attempt frequency of magnetization in nanomagnets with thin-film geometry

Solving the stochastic Landau-Lifshitz-Gilbert equation numerically, we investigate the effect of the potential landscape on the attempt frequency of magnetization in nanomagnets with the thin-film geometry. Numerical estimates of the attempt frequency are analyzed in comparison with theoretical predictions from the Fokker-Planck equation for the Neel-Brown model. It is found that for a nanomagnet with the thin-film geometry, theoretically predicted values for the universal case are in excellent agreement with numerical estimates.

cond-mat.mtrl-sci↗

arXiv2008

Current-induced domain wall motion in a nanowire with perpendicular magnetic anisotropy

We study theoretically the current-induced magnetic domain wall motion in a metallic nanowire with perpendicular magnetic anisotropy. The anisotropy can reduce the critical current density of the domain wall motion. We explain the reduction mechanism and identify the maximal reduction conditions. This result facilitates both fundamental studies and device applications of the current- induced domain wall motion.

cond-mat.mes-hall↗