arXiv subjects

Karthik C. S.

Publications and source records attributed to Karthik C. S..

At least 19 recordsLinked to original sources

arXiv2026

Improved Multilayered PCPs and Hypergraph Vertex Cover

We present two elementary constructions of multilayered PCPs that improve upon prior constructions in two ways. Specifically, we give one construction of quasi-linear size, and another one with $2$-to-$2$ constraints. Using these constructions we obtain the following results for the hypergraph vertex cover problem: $\bullet$ For $k=3$, for all $\varepsilon>0$, approximating the minimum vertex cover of a given $3$-uniform hypergraph within factor $1+\sqrt{2}-\varepsilon$ is NP-hard. Previously, the best known result due to [Dinur, Guruswami, Khot, Regev, SICOMP 2005] achieved a factor of $2-\varepsilon$. $\bullet$ For $k\geq 4$, for all $\varepsilon>0$, approximating the minimum vertex cover of a given $k$-uniform hypergraph within factor $k-\varepsilon$ is NP-hard, which is tight. Previous works established this result assuming the Unique-Games Conjecture [Khot, Regev, JCSS 2008], and a weaker factor of $k-1-\varepsilon$ for standard NP-hardness [Dinur, Guruswami, Khot, Regev, SICOMP 2005]. $\bullet$ Assuming the Exponential Time Hypothesis, for all $k\geq 3$ and $\varepsilon>0$ there is $C>0$ such that no $2^{n/\log^C n}$-time algorithm approximates the minimum vertex cover in a $k$-uniform, $n$-vertex hypergraph within factor $k-1-\varepsilon$. The proofs were obtained using ChatGPT 5.6 Pro and subsequently rewritten by the communicators.

cs.CC↗

arXiv2026

Constant Rate Isometric Embeddings of Hamming Metric into Edit Metric

A function $φ:\{0,1\}^n \to \{0,1\}^N$ is called an isometric embedding of the $n$-dimensional Hamming metric space to the $N$-dimensional edit metric space if, for all $x,y\in\{0,1\}^n$, the Hamming distance between $x$ and $y$ is equal to the edit distance between $φ(x)$ and $φ(y)$. The rate of such an embedding is defined as the ratio $n/N$. It is well known in the literature how to construct isometric embeddings with rate $Ω(1/\log n)$. However, achieving even near-isometric embeddings with positive constant rate has remained elusive until now. In this paper, we present an isometric embedding with rate $1/8$ by discovering connections to synchronization strings, which were studied in the context of insertion-deletion codes (Haeupler-Shahrasbi [JACM'21]). At a technical level, we introduce a framework for obtaining high-rate isometric embeddings using a novel object called misaligners. As an immediate consequence of our constant-rate isometric embedding, we improve known conditional lower bounds for various optimization problems in the edit metric, now with optimal dependence on the dimension. We complement our results by showing that no isometric embedding $φ:\{0,1\}^n \to \{0,1\}^N$ can have rate greater than $15/32$ for all positive integers $n$. En route to proving this upper bound, we uncover fundamental structural properties necessary for every Hamming-to-edit isometric embedding. We also prove similar upper and lower bounds for embeddings over larger alphabets. Finally, we consider embeddings $φ:Σ_{\mathrm{in}}^n \to Σ_{\mathrm{out}}^N$ between different input and output alphabets, where the rate is given by $\frac{n\log|Σ_{\mathrm{in}}|}{N\log|Σ_{\mathrm{out}}|}$. In this setting, we show that the rate can be made arbitrarily close to $1$.

cs.DM↗

arXiv2026

Fine-Grained Complexity of Continuous Euclidean k-Center

In the (continuous) Euclidean $k$-center problem, given $n$ points in $\mathbb{R}^d$ and an integer $k$, the goal is to find $k$ center points in $\mathbb{R}^d$ that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. $\bullet$ Parameterized by $k$: Assuming the Exponential Time Hypothesis (ETH), we show that there is no $f(k)n^{o(k^{1-1/d})}$-time algorithm for the Euclidean $k$-center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any $(1+\varepsilon)$-approximation algorithm running in time $(k/\varepsilon)^{o(k^{1-1/d})}n^{O(1)}$, thereby establishing near-optimality of the corresponding approximation scheme by the same authors. $\bullet$ Small $k$: Assuming the 3-SUM hypothesis, we prove that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the Euclidean $2$-center problem in $\mathbb{R}^3$. This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any $\varepsilon > 0$, the Euclidean $6$-center problem in $\mathbb{R}^2$ also admits no $O(n^{2-\varepsilon})$-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

cs.CG↗

arXiv2026

Near-Optimal Bounds for Parameterized Euclidean k-means

The $k$-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find $k$ representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of $k$-means in Euclidean spaces parameterized by the number of clusters, $k$. In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a $(1+\varepsilon)$-approximation for high-dimensional Euclidean $k$-means in time $2^{(k/\varepsilon)^{O(1)}} \cdot dn^{O(1)}$. In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean $k$-means by showing that there is no $2^{(k/\varepsilon)^{1-o(1)}} \cdot n^{O(1)}$ time algorithm achieving a $(1+\varepsilon)$-approximation for $k$-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is $2^{\tilde{O}(k/\varepsilon)} + O(ndk)$. Furthermore, assuming XXH, we show that the seminal $O(n^{kd+1})$ runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for $k$-means is optimal for small values of $k$.

cs.CG↗

arXiv2026

Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces

The $k$-median and $k$-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the $k$-median (resp. $k$-means) problem is to find $k$ representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a $(1+\varepsilon)$-factor approximation in low-dimensional Euclidean metric for both the $k$-median and $k$-means problems in near-linear time $2^{(1/\varepsilon)^{O(d^2)}} n \cdot \text{polylog}(n)$ (where $d$ is the dimension and $n$ is the number of input points). We improve this running time to $2^{\tilde{O}(1/\varepsilon)^{d-1}} \cdot n \cdot \text{polylog}(n)$, and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no $2^{{o}(1/\varepsilon^{d-1})} n^{O(1)}$ algorithm achieving a $(1+\varepsilon)$-approximation for $k$-means.

cs.CG↗

arXiv2026

Accelerating Scientific Research with Gemini: Case Studies and Common Techniques

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

cs.CL↗

arXiv2026

Impossibility of Depth Reduction in Explainable Clustering

Over the last few years Explainable Clustering has gathered a lot of attention. Dasgupta et al. [ICML'20] initiated the study of explainable $k$-means and $k$-median clustering problems where the explanation is captured by a threshold decision tree which partitions the space at each node using axis parallel hyperplanes. Recently, Laber et al. [Pattern Recognition'23] made a case to consider the depth of the decision tree as an additional complexity measure of interest. In this work, we prove that even when the input points are in the Euclidean plane, then any depth reduction in the explanation incurs unbounded loss in the $k$-means and $k$-median cost. Formally, we show that there exists a data set $X\subseteq \mathbb{R}^2$, for which there is a decision tree of depth $k-1$ whose $k$-means/$k$-median cost matches the optimal clustering cost of $X$, but every decision tree of depth less than $k-1$ has unbounded cost w.r.t. the optimal cost of clustering. We extend our results to the $k$-center objective as well, albeit with weaker guarantees.

cs.LG↗

arXiv2025

Range Longest Increasing Subsequence and its Relatives

In this work, we present a plethora of results for the range longest increasing subsequence problem (Range-LIS) and its variants. The input to RLIS is a sequence $S$ of $n$ real numbers and a collection $Q$ of $m$ query ranges, and for each query in $Q$, the goal is to report the LIS of the sequence $S$ restricted to that query. Our two main results are for the following generalizations of the RLIS problem. 2D range queries: In this variant of the RLIS problem, each query is a pair of ranges, one of indices and the other of values, and we provide a randomized algorithm with running time $\tilde{O}(m n^{1/2} + n^{3/2}) + O(k)$, where $k$ is the cumulative length of the $m$ output subsequences. This improves on the elementary $O(mn)$-time algorithm when $m$ is at least $n^{1/2}$. Previously, the only known result breaking the quadratic barrier was due to Tiskin [SODA'10], which could only handle 1D range queries (i.e., each query was a range of indices) and also just outputted the length of the LIS (instead of reporting the subsequence achieving that length). Colored sequences: In this variant of the RLIS problem, each element in $S$ is colored, and for each query in $Q$, the goal is to report a monochromatic LIS contained in the sequence $S$ restricted to that query. For 2D queries, we provide a randomized algorithm for this colored version with running time $\tilde{O}(m n^{2/3} + n^{5/3}) + O(k)$. Moreover, for 1D queries, we provide an improved algorithm with running time $\tilde{O}(m n^{1/2} + n^{3/2}) + O(k)$. Thus, we again improve on the elementary $O(mn)$-time algorithm. Additionally, assuming the well-known Combinatorial Boolean Matrix Multiplication Hypothesis, we prove that the running time for 1D queries is essentially tight for combinatorial algorithms.

cs.DS↗

arXiv2025

On Inapproximability of Reconfiguration Problems: PSPACE-Hardness and some Tight NP-Hardness Results

Recently, Ohsaka [STACS'23] put forth the Reconfiguration Inapproximability Hypothesis (RIH), which roughly asserts that there is some $ε>0$ such that given as input a $k$-CSP instance (for some constant $k$) over some constant sized alphabet, and two satisfying assignments $ψ_s$ and $ψ_t$, it is PSPACE-hard to find a sequence of assignments starting from $ψ_s$ and ending at $ψ_t$ such that every assignment in the sequence satisfies at least $(1-ε)$ fraction of the constraints and also that every assignment in the sequence is obtained by changing its immediately preceding assignment (in the sequence) on exactly one variable. Assuming RIH, many important reconfiguration problems have been shown to be PSPACE-hard to approximate by Ohsaka [STACS'23; SODA'24]. In this paper, we provide a proof of RIH. Our proof uses known constructions of PCP of Proximity to create the gap, and further leverages a parallelization framework from recent parameterized inapproximability results to analyze the quantitative trade-off between $ε$ and $k$ in RIH. We note that Hirahara and Ohsaka [STOC'24] have also independently proved RIH. We also prove that the aforementioned $k$-CSP Reconfiguration problem is NP-hard to approximate to within a factor of $1/2 + ε$ (for any $ε>0$) when $k=2$. We complement this with a polynomial time $(1/2 - ε)$-approximation algorithm, which improves upon a $(1/4 - ε)$-approximation algorithm of Ohsaka [2023] (again for any $ε>0$). Finally, we show that Set Cover Reconfiguration is NP-hard to approximate to within a factor of $2 - ε$ for any constant $ε> 0$, which matches the simple linear-time 2-approximation algorithm by Ito et al. [TCS'11].

cs.CC↗

arXiv2025

Hardness of Median and Center in the Ulam Metric

The classical rank aggregation problem seeks to combine a set X of n permutations into a single representative "consensus" permutation. In this paper, we investigate two fundamental rank aggregation tasks under the well-studied Ulam metric: computing a median permutation (which minimizes the sum of Ulam distances to X) and computing a center permutation (which minimizes the maximum Ulam distance to X) in two settings. $\bullet$ Continuous Setting: In the continuous setting, the median/center is allowed to be any permutation. It is known that computing a center in the Ulam metric is NP-hard and we add to this by showing that computing a median is NP-hard as well via a simple reduction from the Max-Cut problem. While this result may not be unexpected, it had remained elusive until now and confirms a speculation by Chakraborty, Das, and Krauthgamer [SODA '21]. $\bullet$ Discrete Setting: In the discrete setting, the median/center must be a permutation from the input set. We fully resolve the fine-grained complexity of the discrete median and discrete center problems under the Ulam metric, proving that the naive $\widetilde{O}(n^2 L)$-time algorithm (where L is the length of the permutation) is conditionally optimal. This resolves an open problem raised by Abboud, Bateni, Cohen-Addad, Karthik C. S., and Seddighin [APPROX '23]. Our reductions are inspired by the known fine-grained lower bounds for similarity measures, but we face and overcome several new highly technical challenges.

cs.DS↗

arXiv2025

On Approximability of $\ell_2^2$ Min-Sum Clustering

The $\ell_2^2$ min-sum $k$-clustering problem is to partition an input set into clusters $C_1,\ldots,C_k$ to minimize $\sum_{i=1}^k\sum_{p,q\in C_i}\|p-q\|_2^2$. Although $\ell_2^2$ min-sum $k$-clustering is NP-hard, it is not known whether it is NP-hard to approximate $\ell_2^2$ min-sum $k$-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the $\ell_2^2$ min-sum $k$-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than $1.056$ and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a nearly linear time parameterized PTAS for $\ell_2^2$ min-sum $k$-clustering running in time $O\left(n^{1+o(1)}d\cdot \exp((k\cdot\varepsilon^{-1})^{O(1)})\right)$, where $d$ is the underlying dimension of the input dataset. Finally, we consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label $i\in[k]$ for input point, thereby implicitly partitioning the input dataset into $k$ clusters that induce an approximately optimal solution, up to some amount of adversarial error $α\in\left[0,\frac{1}{2}\right)$. We give a polynomial-time algorithm that outputs a $\frac{1+γα}{(1-α)^2}$-approximation to $\ell_2^2$ min-sum $k$-clustering, for a fixed constant $γ>0$.

cs.DS↗

arXiv2025

Almost Polynomial Factor Inapproximability for Parameterized k-Clique

The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an algorithm runs in time T(k)poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for the k-Clique problem. Although, the non-existence of an F(k)-FPT-approximation algorithm for any computable sublinear function F is known under gap-ETH [Chalermsook et al., FOCS 2017], it has remained a long standing open problem to prove the same inapproximability result under the more standard and weaker assumption, W[1]$\neq$FPT. In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k)=O(1)) FPT-approximation algorithms under W[1]$\neq$FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every $F(k)=k^{1/H(k)}$ factor FPT-approximation algorithm for any increasing computable function H (for example $H(k)=\log^\ast k$). Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin.

cs.CC↗

arXiv2024

On Equivalence of Parameterized Inapproximability of k-Median, k-Max-Coverage, and 2-CSP

Parameterized Inapproximability Hypothesis (PIH) is a central question in the field of parameterized complexity. PIH asserts that given as input a 2-CSP on $k$ variables and alphabet size $n$, it is W[1]-hard parameterized by $k$ to distinguish if the input is perfectly satisfiable or if every assignment to the input violates 1% of the constraints. An important implication of PIH is that it yields the tight parameterized inapproximability of the $k$-maxcoverage problem. In the $k$-maxcoverage problem, we are given as input a set system, a threshold $τ>0$, and a parameter $k$ and the goal is to determine if there exist $k$ sets in the input whose union is at least $τ$ fraction of the entire universe. PIH is known to imply that it is W[1]-hard parameterized by $k$ to distinguish if there are $k$ input sets whose union is at least $τ$ fraction of the universe or if the union of every $k$ input sets is not much larger than $τ\cdot (1-\frac{1}{e})$ fraction of the universe. In this work we present a gap preserving FPT reduction (in the reverse direction) from the $k$-maxcoverage problem to the aforementioned 2-CSP problem, thus showing that the assertion that approximating the $k$-maxcoverage problem to some constant factor is W[1]-hard implies PIH. In addition, we present a gap preserving FPT reduction from the $k$-median problem (in general metrics) to the $k$-maxcoverage problem, further highlighting the power of gap preserving FPT reductions over classical gap preserving polynomial time reductions.

cs.CC↗

arXiv2024

On connections between k-coloring and Euclidean k-means

In the Euclidean $k$-means problems we are given as input a set of $n$ points in $\mathbb{R}^d$ and the goal is to find a set of $k$ points $C\subseteq \mathbb{R}^d$, so as to minimize the sum of the squared Euclidean distances from each point in $P$ to its closest center in $C$. In this paper, we formally explore connections between the $k$-coloring problem on graphs and the Euclidean $k$-means problem. Our results are as follows: $\bullet$ For all $k\ge 3$, we provide a simple reduction from the $k$-coloring problem on regular graphs to the Euclidean $k$-means problem. Moreover, our technique extends to enable a reduction from a structured max-cut problem (which may be considered as a partial 2-coloring problem) to the Euclidean $2$-means problem. Thus, we have a simple and alternate proof of the NP-hardness of Euclidean 2-means problem. $\bullet$ In the other direction, we mimic the $O(1.7297^n)$ time algorithm of Williams [TCS'05] for the max-cut of problem on $n$ vertices to obtain an algorithm for the Euclidean 2-means problem with the same runtime, improving on the naive exhaustive search running in $2^n\cdot \text{poly}(n,d)$ time. $\bullet$ We prove similar results and connections as above for the Euclidean $k$-min-sum problem.

cs.CG↗

arXiv2024

Conditional lower bounds for sparse parameterized 2-CSP: A streamlined proof

Assuming the Exponential Time Hypothesis (ETH), a result of Marx (ToC'10) implies that there is no $f(k)\cdot n^{o(k/\log k)}$ time algorithm that can solve 2-CSPs with $k$ constraints (over a domain of arbitrary large size $n$) for any computable function $f$. This lower bound is widely used to show that certain parameterized problems cannot be solved in time $f(k)\cdot n^{o(k/\log k)}$ time (assuming the ETH). The purpose of this note is to give a streamlined proof of this result.

cs.CC↗

arXiv2024

Inapproximability of Maximum Diameter Clustering for Few Clusters

In the Max-k-diameter problem, we are given a set of points in a metric space, and the goal is to partition the input points into k parts such that the maximum pairwise distance between points in the same part of the partition is minimized. The approximability of the Max-k-diameter problem was studied in the eighties, culminating in the work of Feder and Greene [STOC'88], wherein they showed it is NP-hard to approximate within a factor better than 2 in the $\ell_1$ and $\ell_\infty$ metrics, and NP-hard to approximate within a factor better than 1.969 in the Euclidean metric. This complements the celebrated 2 factor polynomial time approximation algorithm for the problem in general metrics (Gonzalez [TCS'85]; Hochbaum and Shmoys [JACM'86]). Over the last couple of decades, there has been increased interest from the algorithmic community to study the approximability of various clustering objectives when the number of clusters is fixed. In this setting, the framework of coresets has yielded PTAS for most popular clustering objectives, including k-means, k-median, k-center, k-minsum, and so on. In this paper, rather surprisingly, we prove that even when k=3, the Max-k-diameter problem is NP-hard to approximate within a factor of 1.5 in the $\ell_1$-metric (and Hamming metric) and NP-hard to approximate within a factor of 1.304 in the Euclidean metric. Our main conceptual contribution is the introduction of a novel framework called cloud systems which embed hypergraphs into $\ell_p$-metric spaces such that the chromatic number of the hypergraph is related to the quality of the Max-k-diameter clustering of the embedded pointset. Our main technical contributions are the constructions of nontrivial cloud systems in the Euclidean and $\ell_1$-metrics using extremal geometric structures.

cs.CG↗

arXiv2023

On Steiner Trees of the Regular Simplex

In the Euclidean Steiner Tree problem, we are given as input a set of points (called terminals) in the $\ell_2$-metric space and the goal is to find the minimum-cost tree connecting them. Additional points (called Steiner points) from the space can be introduced as nodes in the solution. The seminal works of Arora [JACM'98] and Mitchell [SICOMP'99] provide a Polynomial Time Approximation Scheme (PTAS) for solving the Euclidean Steiner Tree problem in fixed dimensions. However, the problem remains poorly understood in higher dimensions (such as when the dimension is logarithmic in the number of terminals) and ruling out a PTAS for the problem in high dimensions is a notoriously long standing open problem (for example, see Trevisan [SICOMP'00]). Moreover, the explicit construction of optimal Steiner trees remains unknown for almost all well-studied high-dimensional point configurations. Furthermore, a vast majority the state-of-the-art structural results on (high-dimensional) Euclidean Steiner trees were established in the 1960s, with no noteworthy update in over half a century. In this paper, we revisit high-dimensional Euclidean Steiner trees, proving new structural results. We also establish a link between the computational hardness of the Euclidean Steiner Tree problem and understanding the optimal Steiner trees of regular simplices (and simplicial complexes), proposing several conjectures and showing that some of them suffice to resolve the status of the inapproximability of the Euclidean Steiner Tree problem. Motivated by this connection, we investigate optimal Steiner trees of regular simplices, proving new structural properties of their optimal Steiner trees, revisiting an old conjecture of Smith [Algorithmica'92] about their optimal topology, and providing the first explicit, general construction of candidate optimal Steiner trees for that topology.

math.CO↗

arXiv2023

Clustering Categorical Data: Soft Rounding k-modes

Over the last three decades, researchers have intensively explored various clustering tools for categorical data analysis. Despite the proposal of various clustering algorithms, the classical k-modes algorithm remains a popular choice for unsupervised learning of categorical data. Surprisingly, our first insight is that in a natural generative block model, the k-modes algorithm performs poorly for a large range of parameters. We remedy this issue by proposing a soft rounding variant of the k-modes algorithm (SoftModes) and theoretically prove that our variant addresses the drawbacks of the k-modes algorithm in the generative model. Finally, we empirically verify that SoftModes performs well on both synthetic and real-world datasets.

cs.LG↗