Angle K-Means
Authors: Shenfei Pei
Ruiyu Huang
Yiqing Hu
Zhongqi Lin
Xudong Jiang
Zengwei Zheng
Venue: ICLR 2026 Poster (2026)
Keywords: Clustering
K-Means
Accelerate
Angle
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: Fast K-Means via Enhanced Triangle Inequality

Abstract: We propose an accelerated exact $k$-means algorithm, Angle $k$-means.
As its name suggests, the algorithm mainly leverages angular relationships between data points and cluster centers 
to reduce computational overhead. Although grounded in straightforward geometric principles, 
it delivers substantial performance improvements in empirical evaluations.
In contrast to existing acceleration techniques, our model introduces no new hyperparameters,
preserving full compatibility with standard $k$-means.
Theoretical analysis shows that Angle $k$-means maintains linear time complexity 
with respect to both sample size and dimensionality, 
while empirical evaluations on diverse real-world datasets demonstrate 
significant speedup over state-of-the-art algorithms such as ball $k$-means and Exp-ns.

Published as a conference paper at ICLR 2026
ANGLE K-MEANS
Shenfei Pei', Ruiyu Huang', Yiqing Hu', Zhongqi Lin', Xudong Jiang", Zengwei Zheng! Hangghhuu City University, China
NNayyang Technological University, Singapore
seannfeipe@@gmaiii com, 3230 323033sstu.hcc..du
htt.:/223sstuhhcu.eeeo 323030238stu- hzcu. rrzcu.edu.cm. linzg@hzcu, edu,
EXDJiangentu. www.:ntttuuuuuuuoooooo.oo edu
zhengzw@bzcu. eduuueddu.coo cn
ABSTRACT
We propose an accelerated exact k-means algorithm, Angle k-means. As its name suggests, the algorithm mainly leverages angular relationships between data points and cluster centers to reduce computational Cerrhea..Althous grounded in straightforward geometric principles, it delivers substantial performance improve- ments in empirical evaluations. model introduces new hyperparameters, preserving full compatibility with standard m-means. Theoretical analysis shows that Angle k-means maintains linear time complexity with respect both sample size and dimensionality, while em- pirical evaluations on diverse real-world datasets demonstrate significant speedup over state-of-the afrt algorithms such as ball k-means and Exp-ns.
contrast to existing acceleration techniques,
INTRODUCTION
Clustering is fundamental task in machine learning and continues to receive increasing attention. Due to its unsupervised nature, clustering algorithms are frequently employed to process large-scale and unlabeled data samples which can be easily collected from the Internet.
For example, in industrial IoT networks, clustering are used to improve the energy management of wireless sensors, thereby extending network lifespan and reducing energy consumption (Saeedi
al. 2024)). Besides this, various clustering algorithms have been proposed recently and applied in
many fields, such document classification (Dodda & Babu, 2024 Diallo et al. [022; Liu et al.
2025 power systems (Miraftabzadeh et al.) (023). urban global positioning system (Ran et al.
2024
and emotion categorization (Zhou et al. 2023 Salloum et al. 2024).
Although numerous clustering algorithms have emerged recent years, Lloyd's k-means algorithm
[Looyd 1982) remains extensively utilized due to its computational efficiency and conceptual sim-
plicity. The algorithm iterates through three fundamental steps:
Initialization: Select initial cluster centers
Assignment: Assign each sample to nearest cluster center
Update: Recompute cluster centroids based current assignments
Iterate "Assignment" and "Update" steps until convergence
Two primary limitations affect standard implementations: (1) Performance exhibits high sensitivity random center initialization, and (2) The assignment step requires calculating all sample-center distances, becoming computationally prohibitive for large datasets. This work specifically targets acceleration of the assignment step by reducing required distance computations.
Our principal contributions are:
We propose Angle k-means, accelerated exact k-means variant. Through precomputation of geometric auxiliary variables (including the distance and angles between cluster centers), the center closest the sample can be found without calculating the distance between the sample and all cluster centers.
"Corresponding author.

---

Published as a conference paper at ICLR 2026
No hyper- syper-parameters are introduced in our model and it enjoys the same usage the stan- dard k-means, which makes our algorithm more user-friendly.
Comprehensive evaluation across 14 real-worid datasets demonstrates significant speedups over Mate-of-the-ar alternatives (including Ball k-means and Exp-ns) while maintaining identical clustering accuracy.
RELATED WORK
This section reviews accelerated exact k-means algorithms, focusing primarily on tree-based and boundary-based acceleration techniques. We also briefly discuss relevant approximate methods.
Tree-based acceleration: Kanungo et al. (Kanungo et al. 2002) introduced a filtering algorithm where samples are stored in static kd-tree. Samples within the same tree node share candidate cen- ters, avoiding tree reconstruction during center updates. Building on cover tree foundations, Yang et al. (Lang & Schubert 2023) demonstrate that ball covers in cover trees produce tighter bounds than the bounding boxes used in k-d tree approaches. Their proposed Cover means algorithm achieves robust performance across wider parameter ranges compared to traditional tree-based methods.
For scenarios with large cluster counts, Curtin et al. (Curtin 2017) proposed dual-tree k-means where both data and centers are stored in cover trees. Under certain assumptions, this achieves O(n + k log k) per-iteration complexity and is agnostic to tree structure. Despite these advances, tree-based methods exhibit degraded performance in high-dimensional spaces due to the curse of dimensionality.. See (Aljabbouli et al. 2020 Wang et al. 2020) for related algorithms.
Bound-based methods: Elkan's algorithm (Elann 2003) uses triangle inequality to maintain lower bounds per sample, incurring (Ink) memory overhead that scales poorly with cluster count. Ding et al (Ding et al. 2015) proposed the Yinyang k-means algorithm, which not only clusters samples but also centers. When clustering the centers, the standard k-means algorithm is used, and the number of clusters is hyper-parameter The author suggests that its value should not be greater than k/10. It is worth noting that this method not only accelerates the step of "Assignment" but also accelerates the step of "Update".
Hamerly (Hamerly 2010) presented modification and simplification version of Elkan's k-means algorithm. It does not store the lower bounds for all sample-center combinations, so it does not require O(nk) memory. In fact, the total space complexity of this algorithm is O(n). Drake et al. (Drake & Hamerly 2012) combined Hamerly's algorithm with Elkan's is algorithm and proposed Adaptive k-means. In Elkan's and Hamerly's s algorithms, each sample has k and lower bound, respectively, while in adaptive k-means, each sample has b (1 < b < k) lower bounds, where b hyperparameter, that's why it's called adaptive k-means. Annulus k-means (Drake) (013) can be regarded as variant of Hamerly's ss algorithm. Given sample x, Annulus k-means finds the candidate centers by ||x||| - ||||||| S ||x - cl|2. where c is the nearest center previously assigned to and then finds the center closest to x among these candidate centers. Because the region where the candidate centers are located is a ring. is called Annulus k-means. Exponion algorithm (Newling & Fleuret 2016) is also variant of Hamerly's algorithm. Unlike Annulus k-means, Exponion looks for candidate centers a sphere with center c instead of the coordinate origin for sample x, which results in 22 distances be considered by Exponion. In addition, (Newling & Fleuret 2016), the authors also proposed a technique to make the bounds tighter, which can be applied to all boundary- based algorithms. Based on this technique, the author proposed improved version of Exponion, known as Exp-ns. Xia et al. (Xia et al., (202)) proposed Ball k-means and provided both ring and no- ring versions. It characterizes each cluster with a ball and defined the concept of cluster neighbors. For each sample, it finds the cluster center closest to the sample among the cluster neighbors. In addition, the author also defined a stable region for each cluster, and assignments of samples in the stable region will not be changed. Ismkhan et al. (Ismkhan & Izadi (022) proposed KM-G* an accelerated k-means variant that utilizes geometric lower bounds to reduce distance computations. For more related algorithms, please refer to (Cohen-Addad et al. 2020 Hirschberger et al. 2021 Zhang et al. 2024).
Approximate k-means: Deng et al. (Deng & Zhao 2018) proposed fast k-means variant using a k-ncarest neighbor graph, where the set of candidate centers is composed of the nearest centers assigned to x's neighbors previously, for sample x. Usually, the number of nearest neighbors is
2

---

Published as a conference paper at ICLR 2026
smaller than the number of clusters, it has low computational complexity. A fast heuristic method is proposed (Khandelwal & Awekar| (017). where only a subset of the nearby clusters considered as candidates to update the labels of the sample. Grandoni et al. (Grandoni et al. 2022 developed
a refined approximation method for the Fuclidean k-means, which contributes to more precise clus- tering results. More algorithms be found here (Wen et al., [022 Du al.
2024). On improving
the quality of clustering, the Self-Constrained Clustering Ensemble (Wei et al. 2025 integrates self-
supervision and metric learning to iteratively refine base clusterings and improve ensemble quality,
3
OUR MODEL
In this section, we first review the k-means problem, and then we introduce optimization algorithm solve it, called Angle k-means (AKM). Our algorithm can be seen as a fast version of Lloyd's algorithm, so Lloyd's s algorithm will also be briefly reviewed. To introduce AKM, some definitions and lemmas are given first, then a special case of AKM given introduce the model motivation.
Let X = [x..... xx,]'' E Raxd denote the dataset to be clustered, where n and d denote the number of samples and features respectively, y E R the labels of the samples (i.e, y/ = 9 means that x; belongs to the g-th cluster), C [c1.-.' "c&]" E Rexd the centers of clusters, where k denotes the number of clusters, which is a positive integer known in advance, the k-means problem can be formulated as follows:
min
||::
co
9 A
where Ag = (il 1 -). 1 Σ EiEA, X;
To solve problem (1) many efficient algorithms have been proposed, among which the Lloyd's algorithm proposed in [loyd (1982) is widely used. The main steps involved can be expressed as: 1. Update labels y according to the centers C. 2. Calculate the centers C according to That is
Vi arg min [|x - c,112, 1, no
jE(L.-. b}
Cg = Ag KE
X
4g = 1, k.
(3)
In this paper, we mainly focus on the first step of Lloyd's algorithm. Specifically, by pre-calculating some auxiliary variables, such the distance and angles between centers, etc., the center closest to sample x; can be found without calculating the distance between and all centers.
3.1 ANGLE K-MEANS
For convenience, some lemmas and notations are first given before introducing the AKM algorithm. Lemma 1. p and q are two points in the Euclidean space RR,, rı and 12 are constants greater than then the optimal value of the following problem [r1 r2l-
min
||p
4/12
(4)
s.t.
||p||2 TI, ||//|2 = r2
Proof. See supplementary material.
Theorem 2. a and are two points in the Euclidean space IR"", a a b, C2,02, C2 C3, e4 are four constants greater than O. Let d(p. q) denotes the optimal value of the following problem
min
||p q112
p.q
|p al = -1, |p bl = 62
(5)
s.t.
||q all = c3, q bll
we have
d(p, q) / 0 + o Dezes cos(O1 - 82)
(6)
where 0, /abp, 02 /abq.
3

---

Published as a conference paper at ICLR 2026
Proof. Since the operations of rotation and translation are both distance distance-preserving transformations, we assume that b = 0, a = (-r, 0, ... ,0), where r = -la bl| a constant. So, we have:
P1 -E2 cos 01
91 E1 cos 02 92 +.... qh = e3 sin 0g
and
p1 ... + p o sin 01
Let p = (P2,' 1M .Pa), a= (42, .. In.. we have
Pp E _ (pi -1 + ||p q
all
According to Lemma it's easy to find the minimum Euclidean distance between p and q.
In order show our algorithm more clearly, some notations used in this section are introduced first.
1. r [r.-... ,ra"" E R", where >> 0 represents the Euclidean distance between X, and the center the cluster to which belongs. i.e., r; ||x; - Cyll2. where ! Vi- BB] E R", where 3; the angle formed by ocg and xy i.e., Br Zoc 3. ① 101.. 0x]" E R where f,y denotes the angle formed by line oc, and c,cj.
3 [8.. B
The core idea of this paper is to reduce the number of candidate centers, for each sample. Specifi- cally, some centers that are far away from the sample are excluded by some computations with lower complexity.
Given x,, denote y/s as 9. combined with the above symbols, we consider the following two ques- tions:
Case 1: If [l cyll 2 rı. and |3; - 0,i 2 π T/3. Does ||x; Cj 2 r holds?
Case 2: If [| cyll < rú. and |3; - 0,y 2 π T/2. Does ||x; C3 2 T: holds?
The answer to both questions is yes. Next, we give a proof for Case 1. The proof of Problem is omitted here because it is very similar to that of Problem 1.
From Theorem
2 we know that the square of the smallest Euclidean distance between x; and сj is
|x: c1 = llc, e2 2r,l/c, cıll2 cos(B, - -0g))
Note that 3 law of cosines.
0g) is not Cy. This equation is derived from Theorem rather than from the
Since cB, (g)) 1/2, we have
xx - c,112 2 r? |lc, - -,112 -r:lc, c2112
(9)
Since ||c, - c;113 2 ri, we have
|x:
1,1
o Icc, c,lla(lle, csll2 r:)) r?
(10)
A schematic of AKM is appears in Figure 1(b) The points located in the cyan region (case 1) and the purple region (case 2) can be safely ignored when updating the label of x, since their distance to x; exceeds Ti.
Corollary 2.1. For any ICg c,lla 2 0, there exists I we have ||x;
||c,-,,lla such that for cos(B; - 0gi) S t, 2r:
c2l22 Tj-
According to Corollary 2.1 we can judge that some distances between centers and samples are not necessary. Next, we summarize the main steps of Angle k-means algorithm. See Algorithm for details.
Initialization: Caleulate the distances between the samples and the coordinate origin o, u E I...
Update centers: Calculate the cluster centers M according to Eq. as the same as that
in Lloyd's.
4

---

Published as a conference paper at ICLR 2026
Update angles and distances: Calculate the distances matrix D, d,y ||c; C2ll2, the distance between the centers and o, the angles B R² and O 1 according the cluster centers.
Find the set of candidate centers: For sample 9 = Mi, we have
B, {7 cos [3: fgi) / 1, ,k)
(11)
As shown in Figure [[c] the distances between the sample x; and the centers in the blue area do not need to be calculated.
Assignment: For each sample x,, find the nearest center in B; and then update its label.
Note that: When finding candidate centers for x,, we visit these centers in order of their distance from Cg. so that we don't have to visit all the centers.
Time complexity: From Algorithm I is not difficult to know that: It takes O(nD) time to compute the vector Computing the matrix C, D, O vector r, and B takes Onn)), O D), O(k)), O(kD), OnnD), and (n)) time, respectively. Sorting the matrix D takes O log k) time. (mpDD) time is required to update y. where p the mean of the number of candidate centers for each sample, and D is the number of features. We directly denote the complexity of the loop in Line 15 of Algorithm [ by (nk)). Therefore, the total time complexity of per-iteration of the proposed method is O log k + npD + nk), where n, k are the number of samples and clusters, respectively.
Space complexity: In our proposed algorithm, memory is mainly consumed by data matrix X, and variables C. D. and O So the space complexity is O(k2).
We summarize the time and space complexity of the algorithms involved in the experiments, as shown in Table As can be seen, our algorithm comparable these state-of-the-art methods.
4 EXPERIMENTS
In this section, we build experiments on several real-world datasets to verify the performance of the proposed algorithm (the number of distances computed and running time). The rest organized as follows: The datasets used in the experiment are first given, then we introduce the comparison algorithms, and finally, the experimental results are shown.
XL
%
2
21
(a) Triangle inequality
(b) Two special cases
(c) Angle k-means
Figure x, represents the sample that is ready to update its assignment, Cg denotes the center previously assigned to x;, o represents the origin of coordinates. (a): According to the triangle inequality, if [|x; C ||2 < rs, then ||c, - c 2|2 must be less than 2r,, so we only need to consider the center whose distance from cy is less than 2r, (b): Two special cases of Angle k-means: If the centers are located in the cyan or purple areas, then the distance from them to x; must be greater
than equal to 7₁. (c): Angle k-means: if cos( B; 0gj) I ||c, c, 2/ll2/22r), i.c., C₁ is located in the blue area, then we have [Xx, c 2 Tt. So there is no need to calculate the distance between x; and cj.
5

---

Published as a conference paper at ICLR 2026
Algorithm 1: An accelerated exact k-means, Angle h-means
Data: X E R the number of clusters, c
Result: The labels of samples, y
Initialize the labels in random way;
Compute u E IR", U4 = ||x: 2|12, where = the coordinate origin;
while not converge do
Compute the cluster centers, C E paxd by Eq. (3;
Compute v E _
| 2l12 ;D E k du = ||c: c,lla:
Obtain the index matrix H that would sort D, hig = g means c₂ is the j-th neighbor of c;;
Compute E * based on v and D using the law of cosines
Compute R T3
|| Cgll2, where 9 yi
Compute 3 E Rn based on u, v and r using the law of cosines;
10
for 1..................... , do
11
= V/. hg2. B; Ø;
12
dgs 2r; then
Continue:
14
I1 h11 = does not need to be considered;
for j 2, ... k do
16
hgg
17
if dgl >= 2ry then
Break:
cos [8 8gt) > (2r:) then
19
Add set B;
Update y: arg min ||x: c,|12;
22
Output the labels y _=:
4.1 DATASETS
14 real datasets are used, including 10 middle-scale datasets: Digits, Corel5k, Coil100, CNBC, Iso- let, RaFD, USPS, PINS, CPLFW, and FYaleB, as well as 4 large-scale datasets: L-CAS, L-CLBA, L-YTF, and L-EDS. The datasets contain samples ranging from 4000 to 621,126 with dimensions ranging from 256 to 1024. The statistics of the datasets are shown in Table 2 and more detailed information can be found in the appendix.
4.2 BASELINES
Several accelerated exact k-means algorithms are considered, including: Elkan's k-means (Elkan 2003). Annulus k-means (Ann) (Drake) 2013), Exp-ns (Newling & Fleuret 2016). and Ball k-means (Xia et al. 2022). To demonstrate the performance these algorithms, standard h-means (Lloyd
Table Time and space complexity'
Algorithms
Setup'
Time Cost
Space cost
Lloyd's [Loovd 1982
0(1)
O(nkD)
O(k + n)
Elkan's 's bblkan
2003
O(1)
O D knD)
O(k² kn)
Annulus Drake
2013
O(kD nD)
O(kD log nD logk I'D | knD)
O(k" kn)
Exp-ns Newling & Fleuret, 2016
O(1)
O D log + nD log logk k'D | knD)
O(k" kn)
Ball k-mmaas
(Xia et al. 2022
O kmD)
O D km log mn +DD)
O + km)
Angle k-means
O(nD)
O D log k + mpD nk)
O(k2)
D number features. "Setup" refers to the initialization overhead required to accelerate the algorithm. and n' are the average pumber of cluster neighbors and samples the active area.
is the average number of candidate centers for cach sample.

---

Published as a conference paper at ICLR 2026
1982) also considered. Since AKM does not introduce additional parameters, only compare with methods parameter-free
All codes implemented in C/C++, and python interfaces provided. We run them the Ubuntu machine with 64 GB main memory, 5.10 GHz 15-13600 CPU. Source code provided in the supplementary material.
4.3 EXPERIMENTAL RESULTS
As can be seen from Table 3 and 6 and Figure 2 it is clearly seen that:
On most datasets, our algorithm uses the fewest sample-center distances; and as these dis- tance calculations dominate runtime across algorithms. it runs fastest on most datasets. Specifically, on four datasets including RaFD (k=200), EaaleB (k=200), L-YTF (k=2000) and L-CLBA (k=10000), the distance computation our algorithm is only 10.15% 6.7%, 15.5% and 14.81 4.81% %% of that of Lloyd's algorithm respectively, and the running time is only 32.13%, 39. 28%, 38.39% and 52. 16% of that Lloyd's algorithm.
When is larger for a dataset, our algorithm tends have shorter running time and fewer distance calculations. This trend can be observed in datasets such as Corel5k, PINS, L- CLBA and L-CAS.
CONCLUSION
We introduced Angle k-means, parameter-free accelerated exact k-means algorithm that leverages both inter-center distances and angular relationships. By precomputing these geometric proper- ties, we developed an efficient candidate center filtering mechanism with theoretical guarantees. The algorithm maintains computational complexity competitive with distance-only methods since angles derive directly from precomputed distances. Specifically, per-iteration time complexity is O(k log + +npD++ +nk) with O(k²) space overhead, where p is the average number of candidate centers for each sample, and D is the number of the features. Experimental validation across diverse real-world datasets demonstrates significant speed improvements over maae--ofthhe- alternatives. The performance advantage of AKM scales with the number of clusters, becoming increasingly sig- nificant for larger values of k. This aligns with practical large-scale clustering scenarios where a larger sample size typically corresponds to a larger number of centers.
5.1 FUTURE WORK
While current angle computations reference the coordinate origin for convenience, future work will explore angular relationships relative to: 1) Multiple reference points to enhance filtering perfor- mance. 2) Dynamically selected geometric anchors. Such extensions could further reduce distance computations while maintaining theoretical guarantees.
Table Description of the datasets
Datascts
Digits
Corc15k
CNBC
Coil100
Isolet
RaFD
USPS
4000
5000
6574
7200
7797
8040
9298
256
423
256
1024
617
256
256
10
50
236
100
26
67
10
Datasets
PINS
CPLFW
EYaleB
L-CAS
L-CLBA
L-YTF
L-EDS
10770
11652
16128
30863
202599
621 126
240000
256
256
256
256
256
256
784
100
3930
28
1040
10177
1595
10

---

Published as a conference paper at ICLR 2026
Table 3: Per-iteration average of distance computation on middle-scale datasets (% of Lloyd).
Dataset
d
k
Iter
Elkan
Ann
Exp
Ball-R
Ball-noR
Angle
Corel5k
5000
423
30
39
81.43%
97.61% %
82.88%
77.51% %
98.92%
66.81 %
50
66
67.96%
91.13%
68.82%
65.07%
96.60%
53.06%
100
26
55.48%
87.10%
52.84%
51.00%
91.54%
40.60%
200
27
41.53%
76.67%
38.39%
37.50%
80.40%
29.00%
Coil100
7200
1024
30
45
61.80%
72.28%
63.75%
58.62%
93.61%
47.50%
50
39
51.80%
66.57%
51.06%
47.89%
87.92%
39.71%
100
32
39.23%
56.29%
36.83%
35.35%
73.81%
29.84%
200
29
25.87%
44.15%
22.22%
21.57%
57.59%
19.50%
Isolet
7797
617
30
19
87.12%
96.98%
85.64%
80.74%
97.13%
74.56%
50
26
80.57%
96.92%
78.72%
75.41%
96.99%
67.25%
100
39
74.11%
96.44%
72.52%
70.75%
96.46%
59.03%
200
17
65.67%
88.61%
59.97%
59.09%
89.07%
51.99%
RaFD
8040
256
30
16
64.22%
60.80%
57.67%
55.82%
62.35%
53.71%
50
8
61.02%
58.64%
48.78%
46.14%
77.51%
42.85%
100
11
42.83%
57.42%
34.62%
32.91%
64.61%
20.51
200
26
22.28%
50.82%
18.92%
18.12%
57.19%
10.15%
PINS
10770
256
30
56
96.37%
99.64%
96.52%
93.39%
100.12%
83.11%
50
25
94.00%
96.32%
90.58%
88.50%
96.54%
69.79%
100
13
84.96%
91.81%
77.51%
76.33%
91.99%
43.56%
200
20
75.98%
94.65%
71.41%
70.62%
94.68%
29.93%
EYaleB
16128 256
30
27
32.60%
46.59%
28.11%
22.93%
83.48%
15.28%
50
31
21.49%
51.36%
20.48%
17.03%
89.54%
12.42%
100
32
16.95%
51.31%
15.10%
13.24%
78.15%
10.39%
200
59
10.29%
33.06%
9.16%
8.24%
57.71%
6.70%
Table Per-iteration average running time (s) on middle-scale datasets (% of Lloyd).
Dataset
k
Iter
Elkan
Ann
Exp
Ball-R
Ball-noR
Angle
Corel5k
5000
423
30
39
88.51
4927.02%
4831.68%
1365.66%
1609.40%
83.11%
50
66
86.39%
5774.38%
4830.50%
1408.30%
1319.91%
72.60%
100
26
69.70%
6741.17%
4415.10%
1654.41%
1318.73%
54.80%
200
27
60.60%
6686.30%
3631,00%
1593.20%
1506.99%
48.88%
Coil100
7200
1024
30
45
95.04%
5843.50%
6233.21%: %
1801.06%
2065.44%
81.73%
50
39
81.51%
7948.64%
5358.94%
1435.50%
1566.24%
71.48%
100
32
68.62%
6611.39%
4636.20%
1353.69%
1367.39%
68.24%
200
29
60.16%
5972.03%
3372.47%
1317.36%
1306.37%
57.33%
Isolet
7797
617
30
19
74.43%
4653.21%
4821.89%
1281.84%
1414.29%
67.83%
50
26
68.57%
5589.72%
5032.31%
1301.43%
1241.23%
58.75%
100
39
89.58%
9121.26%
7070.51%
1973.01%
1592.17%
78.55%
200
82.44%
8325.37%
5797.87%
2152.18%
1542.48%
66.97%
RaFD
8040
256
30
16
68.70%
1995.86%
2119.69%
515.14%
476.33%
54.83%
50
58.12%
2202.27%
2069.4599
608.71%
588.18%
49.47%
100
11
58.45%
3477.80%
2350.80%
878.46%
694.44%
38.11%
200
26
47.14%
4380.74%
1894.59%
836.19%
886.05%
32.13%
PINS
10770 256
30
56
99.62%
4395.61%
4777.55%
1069.51%
1109.70%
77.77%
50
25
74.17%
4096.01%
4265.10%
839.92%
734.73%
62.39%
100
13
79.10%
5369.81%
4956.40%
1047.10%
791.66%
50.68%
200
20
101.48%
9053.59%
7421.47%
1853.29%
1272.49%
56.14%
EYaleB
16128 256
30
27
50.24%
2274.60%
2012.81%
836.33%
955.72%
41.28%
50
31
50.69%
3413.31%
2049.89%
848.07%
1010.59%
42.87%
100
32
55.47%
4867.13%
2304.21%
733.16%
1131.99%
43.28%
200
59
42.96%
4393.32%
1882.89%
740.65%
999.21%
39.28%
8

---

Published as a conference paper at ICLR 2026
1.4e+06
1.4c+06
number of distances
1.28+06
1.0e+06
1.2e+06
of
8.08+05
06
6.0e+05
0.0e+05
2
10
12
15
10
1.5
20
25
Iteration
Iterationn
(a) Isolet
(b) Coil100
1.0e+06
3.0e+06
number of distances
9.0e+055
number of distances
2.5e+06
8.0e+05
7.0e+00-
2.0e+06
0,0e+05
1.5e+06
5.0e+05-
1.0e+06
0.0e+05
5.0e+05
3.0e+05
20
Iteration
Iteration
(c) Corel5k
(d) EYaleB
2.0e+06
1.5e+06
number of distances
1.2e+06
1.5e+06
1.0e+06
1.2e+06
7.5e+05
1.0e+06
5.0e+05
7.5e+05
2.5e+05
8-0
10
15
10
1.5
20
25
Iteration
Iterattion
(e) PINS
(f) RaFD
3.0e+07
2.0e+09
1.8e+09
Ball nottihh has somte number distance computation
2.5e+07
number of distances
1.5e+09
2.0e+07
1.2e+09
Ball RJB0 has the sartee computatiion
number of
1.5e+07
7.5e+08
1.0e+07
5.0e+08
5.0e+06
2.5e+08
10
20
Iteration
Iteration
(g) L-CAS
(h) I.-LBBA
Figure 2: The number of calculated distances in each iteration on middle-scale datasets and large- scale datasets L-CAS, L-CLBB. Due the large data volume and long computation time, only the number of distance calculations in the first iteration was recorded for Ann, Exp. Ball-R, and Ball- noR on L-CLBA.

---

Published as a conference paper at ICLR 2026
1.2e+09
2.2e+08
number distances
0.0
Bajl Exp. has Syrte computtaton
number of didistnnces
1.0c+09
0.8
6.0e+08
and (xp(e) has the same computation
0.
6.0e+08
1.2e+08
4.0e+08
1.0e+08
2.0e+08
25
50
75
100
125
150
0
20
40
60
80
100
(a) L-EDS
(b) L-YTF
Figure 3: The number of calculated distances in each iteration on large-scale datasets L- L-F and L-EDS. Due the large data volume and long computation time, only the number of distance calculations in the first iteration was recorded for Ann, Exp, Ball-R, and Ball-noR.
Table 5: Per-iteration average of distance computation large-scale datasets (% of Lloyd).
Dataset n
k
Iter
Elkan
Ann
Exp
Ball-R
Ball-noR
Angle
L-CAS
30863
256
300
41
89.05%
92.85%
86.76%
86.38%
95.68%
75.92%
500
29
76.51%
86.65%
73.14%
72.90%
90.27%
57.89%
1000
24
38.92%
70.92%
34.82%
34.67%
74.89%
22.71 %
L-CLBA 202599
256
5000
72
41.84%
54.75%(1)
40.30%(1)
40.58%(3)
27.70% 55.45%(()
8000
39
35.23%
52.55%(()
33.24%(()
33.22%:()
53.31%(3)
18.56%
10000
40
31.38%
50.45%(l)
29.41%(()
29.52%((1
51.33%((1
14.81 %
L-EDS 240000 784
500
137
69.58%
97.49%(1)
72.74%(1)
72.41%(1)
100.08%(1)
48.41 %
800
227
62.55%
96.93%(()
65.84%(() 65.62%(()
100.02%(1
41.69%
1000
144
59.24%
96.67%(1
62.49%(1) 62.31%(1
100.00%(()
38.95%
L-YTF 621126 256
500
246
55.37%
(1.03%((
54.04%(1
54.40%(5)
82.14%(5)
45.03%
1000
136
40.84%
39.95%(1 (1)
40.18%(5)
76.30%(5)
29.93%
2000
100
24.68%
67.72%
23.82%(()
23.75%(1)
70.13%(1)
15.50%
The numbers in parentheses indicate the iteration counts recorded for the corresponding algorithms.
Table 6: Per-iteration average running time (s) on large-scale datasets (% of Lloyd).
Datasets
n
d
k
Iter
Elkan
Ann
Exp
Ball-R
Ball-noR
Angle
L-CAS
30863
256
300
115.79%
10321.4244
10386.03%
1641.33%
1419.43%
92.93%
500
29
108.78%
10589.16%
9879.55%
1937.89%
1510.95%
93.35% %
1000
24 69.36%
9808.129 12%
5260.08%
1995.84%
1779.16%
50.20%
L-CLBA 202599 256
5000
72 91.34%
10000.17%(1)
17011.42%(1
2821.78%(3 2189.28%6(3
67.30%
8000
39 88.95%
11965.33%(1 55878.0880
(1) 55878.09%(1
4705. (705.84%((3 3028. (698.)6 56.56 %
10000
40 85.29%
(0662.26%(()
29391 66%(() 5070.91 (6(1)
52.16%
L-EDS 240000 784
500 137 81.65%
(9795.48%)()
20090.79%6(1 3613.51 (1))) (192))))(() 64.80 %
800 227 112.51% % (3%)).16%(
42984.06% 5228.59%6(1 (675.39%(1 85.03 %
1000 144 104. 104.14%
(0514.76%(
44914.60%
4237.86% 3212.62%(( 79.41 %
L-YTF 621126 256
500 246
104. 14%
13154.24%
(0363.34%(1) 1740.) 1674.65%)() 91.11 %
1000
136 81.35% (1) 9107.6 (107.67%(1)
157).) 1435. (23()) 67.17% %
2000 100 55.19%
11802.59%(1) 1911.91%(1 958,09%(() 1298.8 18..81%6 81%(1) 38.39 %
The numbers in parentheses indicate the iteration counts recorded for the corresponding algorithms.
ACKNOWLEDGMENTS
This work supported in part by the National Natural Science Foundation of China under Grants 62406274, 32401685, and 62576304, in part by the Natural Science Foundation of Zhejiang Province, China under Grants LON25F0200299 and LQ24F020021.
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Hasan Aljabbouli, Abdullah Albizri, and Antoine Harfouche. Tree-based algorithm for stable and efficient data clustering. Informatics, 7(4), 2020.
Vincent Cohen-Addad, Silvio Lattanzi, Ashkan Norouzi-Fard, Christian Sohler, and Ola Svensson. Fast and accurate k-means++ via rejection sampling. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 16235-16245, 2020.
Ryan R. Curtin. A Dual-Tree Algorithm, for Fast k-means Clustering With Large k. pp. 300-308. Society for Industrial and Applied Mathematics, 2017.
Cheng-Hao Deng and Wan-Lei Zhao. Fast k-means based on k-nn graph. In 2018 IEEE 34th international conference on data engineering (ICDE), pp. 1220- 1223, 2018.
Bassoma Diallo, Jie Hu, Tianrui Li. Ghufran Ahmad Khan, and Ahmed Saad Hussein. Multi- view document clustering based on geometrical similarity measurement, International Journal of Machine Learning and Cybernetics, 13(3):663-675 2022.
Yufei Ding, Yue Zhao, Xipeng Shen, Madanlal Musuvathi, and Todd Mytkowicz. Yinyang k-means: A drop-in replacement the classic k-means with consistent speedup. In International conference on machine learning, pp. 579-587, 2015.
Rainam Dodda and A. Suresh Babu. Text document clustering using modified particle swarm op- timization with k-means model. International Journal on Artificial Intelligence Tools, 33(01): 2350061, 2024.
Jonathan Drake. Faster k-means clustering. PhD thesis, Baylor University, 2013.
Jonathan Drake and Greg Hamerly. Accelerated k-means with adaptive distance bounds. In 5th NIPS workshop on optimization for machine learning, 2012.
Liang Du, Yunhui Liang, Mian Ilyas Ahmad, and Peng Zhou. K-means clustering based on cheby- shev polynomial graph filtering. In ICASSP 2024 2024 IEEE International Conference Acoustics, Speech and Signal Processing (ICASSP), pp. 7175-7179, 2024.
Charles Elkan. Using the triangle inequality to accelerate k- k-mes.. In Proceedings of the Twentieth International Conference International Conference on Machine Learning, pp. 147-153, 2003.
Yushu Feng, Huan Wang, Haoji Roland Hu, Lu Yu, Wei Wang, and Shiyan Wang. Triplet distillation for deep face recognition. In 2020 IEEE International Conference on Image Processing (ICIP), pp. 808-812, 2020.
Yuquan Gao, Guoxi Sun, and Xinzhuo Zhao. Single face image super-resolution reconstruction with wasserstein generative adversarial networks. In 2024 IEEE 4th International Conference on Software Engineering and Artificial Intelligence (SEAI), pp. 63-67, 2024.
Fabrizio Grandoni, Rafail Ostrovsky, Yuval Rabani, Leonard J. Schulman, and Rakesh Venkat. A refined approximation for euclidean k-means. Information Processing Letters, 176:106251, 2022.
Greg Hamerly, Making k-means even faster. In Proceedings of the 2010 SIAM International Con- fference Data Mining (SDM), pp. 130-140, 2010.
Florian Hirschberger, Dennis Forster, and Jorg Lucke. A variational acceleration for efficient clustering at very large scales. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021.
Hassan Ismkhan and Mohammad Izadi. K-means-g*: Accelerating k-means clustering algorithm utilizing primitive geometric concepts. Information Sciences, 518:298-316 2022.
Vinoj Jayasundara, Sandaru Jayasekara, Hirunima Jayasekara, Jathushan Rajasegaran, Suranga Seneviratne, and Ranga Rodrigo. Textcaps: Handwritten character recognition with very small datasets. 2019 IEEE Winter Conference on Applications of Computer Vision (WACV), pp- 254-262, 2019.
11

---

Published as a conference paper at ICLR 2026
Zhao Kang, Chong Peng, and Qiang Cheng. Robust subspace clustering via smoothed rank approx- imation. IEEE Signal Processing Letters,
2015.
T. Kanungo, D.M. Mount, N.S. Netanyahu, C.D. Piatko, R. Silverman, and A.Y. Wu. An efficient k- means clustering algorithm: analysis and implementation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(7):881-8)) 2002.
Siddhesh Khandelwal and Amit Awekar. Faster k-means cluster estimation. In Advances Infor- mation Retrieval, pp. 520-526, 2017.
Andreas Lang and Erich Schubert. Accelerating k-means clustering with cover trees. In Similarity Search and Applications, pp. 148-162, 2023.
Jianghan Liu, Ziyu Shang, Wenjun Ke, Peng Wang, Zhizhao Luo, Jiajun Liu, Guozheng Li, and Yining Li. LLM-guided semantic-aware clustering for topic modeling. In Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers),
pp. 18420-18435. Association for Computational Linguistics, 2025.
S. Lloyd. Least squares quantization in pem. IEEE transactions on information theory, 28(2): 137-137, 1982.
Firoz Mahmud, Bayezid Islam, Arfat Hossain, and Pushpen Bikash Goala. Facial region segmenta- tion based emotion recognition using k-nearest neighbors. In 2018 International Conference on Innovation in Engineering and Technology (ICIET), pp. 1-5. 2018.
Seyed Mahdi Miraftabzadeh, Cristian Giovanni Colombo, Michela Longo, and Federica Foiadelli. K-means and alternative clustering methods in modern power systems. IEEE Access, 11:1 19596 119633, 2023.
B Chandra Mohan, T Krishna Chaitanya, and T Tirupal. Fast and accurate content based image classification and retrieval using gaussian hermite moments applied to coil 20 and coil 100. In 2019 10th International Conference Computing, Communication and Networking Technolo- gies (ICCCNT), pp. 1-5, 2019. doi: 10. 109/CCCCT4457002009535 8944775.
James Newling and Francois Fleuret. Fast k-means with accurate bounds. International Confer-
ence on Machine Learning, pp. 936-9996 0666944,20 2016.
Shenfei Pei, Huimmn Chen, Feiping Nie, Rong Wang, and Xuelong Li. Centerless clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(1):167-181, 2023.
Ignacio Ramirez, Pablo Sprechmann, and Guillermo Sapiro. Classification and clustering via dictio- nary learning with structured incoherence and shared features. In 2010 IEEE Computer Society Conference on Computer Vision and Puttern Recognition, pp. 3501-3508, 2010.
Xiaojuan Ran, Naret Suyaroj. Worawit Tepsan, Jianghong Ma, Xiangbing Zhou, and Wu Deng. A hybrid genetic-fuzzy ant colony optimization algorithm for automatic k-means clustering in urban global positioning system. Engineering Applications of Artificial Intelligence, 137:109237, 2024.
Ahmad Sacedi, Marjan Kuchaki Rafsanjani. and Samaneh Yazdani. Energy efficient clustering in iot-based wireless sensor networks using binary whale optimization algorithm and fuzzy inference system. The Journal of Supercomputing. (209), 2024.
Said Salloum, Khadija Alhumaid, Ayham Salloum, and Khaled Shaalan. K-means clustering of tweet emotions: A 2d pca visualization approach. Procedia Computer Science, 244:30-36, 2024.
Sheng Wang, Yuan Sun, and Zhifeng Bao. On the efficiency of k-means clustering: evaluation, optimization, and algorithm selection. Proceedings of the VLDB Endowment, 14(2):133-775, 2020.
Wei Wei, Jianguo Wu, Xinyao Guo, Jing Yan, and Jiye Liang. Self-constrained clustering ensemble. IEEE Transactions Pattern Analysis and Machine Intelligence, pp. 1-15, 2025.
Qing Wen, Junkuan Wang, Yabin Shao, and Zizhong Chen. A fast k-means algorithm based on multi- muntiigraaullarity. In International Conference on Internet of Things and Machine Learning (loTMI. 202/), volume 12174, pp. 230 235, 2022.
12

---

Published as a conference paper at ICLR 2026
Shuyin Xia, Daowan Peng, Deyu Meng. Changqing Zhang, Guoyin Wang, Elisabeth Giem, Wei Wei, and Zizhong Chen. Ball k-means: Fast adaptive clustering with no bounds. IEEE Transactions Pattern Analysis and Machine Intelligence. 44(1):87-99, 2022.
Guangyi Xie, Baolong Guo, Zhe Huang, Yan Zheng, and Yunyi Yan. Combination of dominant color descriptor and hu moments in consistent zone for content based image retrieval. IEEE Access, 8: 146284-146299, 2020.
Haowen Zhang, Jing Li, Junru Zhang. and Yabo Dong. Speeding up k-means clustering in high dimensions by pruning unnecessary distance computations. Knowledge-Bact Based Systems, 284: 111262, 2024.
Qian Zhou, Yifan Lei, Hui Du, et al. Public concerns and attitudes towards autism on chinese social media based on k-means algorithm. Scientific Reports, 13(1):15173, 2023.
A SUPPLEMENTARY MATERIAL FOR ANGLE K-MEANS
A.1 PROOF OF LEMMA
Lemma Let p and q be two points in the Euclidean space IR,, and let r1 and r2 be constants greater than O. Then the optimal value of the following problem is |ri rrl::
min
|p
gll s.t. ||pll2 = rı, ||ll22 r2-
(12)
Proof. Let o 0 denote the center of the two spheres, we have ||p o|12 = Tı and | r2-
cos <poq r + r
|lp
all2
< (r1 121-2a r2)² < |pp E
(13)
2r1r2
So far, it is easy to get: ||p gll2 Ir ral.
|lp gll2 Iri r2 if and only if <poq 0.
A.2 DATASETS
This section presents the sources of the datasets used. A total of 14 real datasets are used. 8 face datasets: CNBC et al.) 2024). PINS, RaFD et al., 2018), CPLFW (Feng et al. 2020). EYaleB(Kang et al. 2015). L- L-CAS al. 2023). L-CLBA(Pei et al. 2023), and L-YTF et al.) et al. 2019), and USPS/Ramires 2023 3 handwritten digits datasets: Digits, L-EDS
al. 2010). 3 other datasets: Coil (o et al.] 2019), Isolet Ramirez et al. 2010) and Corel5k(Xie et al. 20200). The datasets contain samples ranging from 4000 to 621, 126 with
mensions ranging from 256 to 1024.
13

---

Published as a conference paper at ICLR 2026
A.3 EXPERIMENT
Table 7: The detailed statistics for the Angle k-means and Lloyd's algorithms.
Angle k-means
Lloyd's k-means
Datasets
n
Iter
to
t1
72
t2
fak
70g
ta
tim
X
2600
14.34
0.0012
5300
0.004
4.4E+04
0.0049
0.1253
2.60E+05
0.0339
0,4863
(2)
5200
13.54
0.0023
10500
0.0027
8.8F+04
0.0094 0.1626 5.20F+05
0.0645
0.8727
X
7800
13.96
0.0033
15700
0.0022
1.3E+05
0.0147
0.2357
7.80E+05
0.0896
1.2512
X
10400
14.08
0.0044
20900
0.0024
1.8E+05
0.0202
0.3207
1.00E+06
0.1056
1,4866
(()
13000
14.68
0.0055 26100 0.0013
2.2E+05
0.0271 0.4224
1.30B+06
0.1129
1.6573
(
15600
15.04
0.0066 31300 0.0016 2.6E+05
0.0356 0.5627 1.60E+06
0.1284
1.9311
(7)
18200
14.62 0.0078 36500
0.0012 3.1E+05
0.0439
0.6656
1.80F406
0.1372
2.0065
(8)
20800
14.28 0.0089 41700 0.0013
3.5E+05
0.0566
0.8338
2.10E+06
0.1561
2.2285
X
23400
14.54
0.0099
46900
0.0017 4.0E+05
0.0701
1.0512
2.30E+06
0.1668
2.4253
X09
26000
14.62
0.0111
52100
0.002
4.4E+05
0.0807
1.2183
2.60E+06
0. 081881
2.7496
340
6.1
380
600
200
3008
raloe aolec
mg and l2 Figure 4: The relationship between the number of calculated distances and the running time.
(a) no and to
(c) mi and
A.3.1 ANALYSTS OF RUNNING TIME AND DISTANCE CALCULATION RELATIONSHIP
is difficult to find the relationship between the running time and the number of distances by looking at the results different datasets, because the rumning time is related to many variables, such the number of features, the number of clusters, and the number of iterations. Therefore, we augment the
dataset X e dd as follows to obtain (1)),.- ... (10), where x = [X,... .X] + e E Rdxin where e represents the noise matrix, which used to avoid duplicate samples X Ar used 2600, d = 256,
here, this is to say,
100.
We run Lloyd's and angle k-means on these datasets and present the experimental results in Table The meanings of the symbols involved in Tables are as follows:
n is the number of samples in the dataset.
no is the number of distances required the initialization in Angle k-means.
lg the time required for the initialization in Angle k-means.
t the average time spent computing auxiliary variables in each iteration.
n22 is the average number of sample-center distances in Angle k-means, n1 np.
t2 is the average time spent updating assignments and centers in Angle k-means.
Iter represents the number of iterations required for the algorithm to converge.
tak is the total time spent by Angle k-means, fak = to + (t1 12) x Iter.
ng is the average number of sample-center distances in Lloyd's algorithm, n3 = nk
t3 the average time spent updating assignments and centers in Lloyd's algorithm.
dem is the total time spent by Lloyd's algorithm, 4kmm Iter.
14

---

Published as a conference paper at ICLR 2026
From Table 7 and Figure 4
that:
In all cases, t1 is much smaller than 12. In other words, computing the auxiliary variable takes much less time than computing the center-sample distance.
Since lax (12 + 41) x Ile lo. occupies major part of the total time of Angle k-means.
The time used to calculate the senter-sample distance, t2, is approximately linearly related to the number of center-sample distances, f2.-
Based on the above findings, smaller value of p leads to smaller value of ng, which in turn results smaller running time.
A.3.2 FULL RESULTS OF RUNNING TIME AND DISTANCE COMPUTATION
In Tables
5 and of the article, only show part of the results. Here are the full results. From 10 and II we can see that Angle k-means requires fewer distance computations while
Tables 8
maintaining faster running times most datasets.
15

---

Published as a conference paper at ICLR 2026
Table 8: Per-iteration average of distance computation on middle-scale datasets.
Dataset
d
k
Iter
Lloyd
Elkan
Ann
Exp
Ball-R
Ball-noR
Angle
Digits
4000
256
30
32
1.2F+05
1.2F+05
1,2F+05
1.1F+05
1.1++05
1,2F+05
1.0F+05
50
22
2.0F+05
1.9F+05
1,9F+05
1.8E+05
1.7E+05
1.9E+05
1.5E+05
100
4.0E+05
3.4E+05
3.8E+05
3.3E+05
3.2E+05
3.8E+05
2.6E+05
200
19
8.0E+05
5.8E+05
7.3E+05
5.4E+05
5.3E+05
7,4E+05
4.1E+00
Corel5k
5000
423
30
39
1.5E+05
1.2E+05
1.5E+05
1.2E+05
1.2E+05
1.5E+05
1.0E+05
50
66
2.5E+05
1.7E+05
2.3E+05
1.7E+05
1.6E+05
2.4E+05
1.3E+05
100
5.0E+05
2.8E+05
4,4E+05
2.6E+05
2.5E+05
4.6E+05
2.0E+05
200
27
1,0E+06
4.2E+05
7.7E+05
3.8E+05
3.7E+05
8,0E+05
2.9E+05
CNBC
6574
256
30
44
2.0E+05
1.5E+05
1.7E+05
1.5E+05
1.4E+05
1.8E+05
1.4++05
50
35
3.3E+05
2,4E+05
2.7E+05
2,4E+05
2.3E+05
3.0E+05
2.2E+00
100
53
6.6B405
4.1E+05
5.1E+05
4.1E+05
4.0E+05
6,2E+05
3.4E+05
200
23
1.3E406
6.0E+05
9.0E+05
5,5E+05
5.4E+05
1.1E+06
3.9
Coil100
7200
1024
30
45
2.2E+05
1.3E+05
1.6E+05
1.4E+05
1.3E+05
2.0E+05
1.0+055
50
39
3,6E+05
1.9E+05
2,4E+05
1.8E+05
1.7E+05
3,2E+05
1,E++05
100
32
7.2F+05
2.8F+05
4.1
2.7E+05
2.5E+05
5.3
2.1F+05
200
29
1.4E406
3.7E+05
6.4++05
3.2E+05
3.1E+05
8.3F+05
2.8E+05
Isolet
7797
617
30
19
2.3E+05
2.0E+05
2.3E+05
2.0E+05
1.9E+05
2.3E+05
1.7E+05
50
26
3,9E+05
3.1E+05
3,8E+05
3.1E+05
2.9E+05
3.8E+05
2.6E+05
100
39
7,8E+05
5,8E+05
7,5E+05
5.7E+05
5.5E+05
7.5E+05
4.6E+05
200
17
1.6E406
1.0E+06
1,4E+06
9.4E+05
9.2E+05
1,44406
8.1E+05
RaFD
8040
256
30
2.4B+05
1.5E+05
1,5E+05
1.4E+05
1.3E+0S
1,SE+05
1.3E+05
50
4.0E+05
2.5E+05
2,4E+05
2.0E+05
1.9E+05
3,1E+05
1.7E+05
100
11
8,0E+05
3,4E+05
4,6E+05
2,8E+05
2,6E+05
5,2E+05
1.6E+05
200
26
1,6E+06
3.6E+05
8.2E+05
3,0E+05
2.9E+05
9,2E+05
1.6+005
USPS
9298
256
30
32
2.8E+05
1.8E+05
2.5E+05
1.8E+05
1.7E+05
2.5E+05
1.5E+05
50
34
4.6E+05
2.7E+05
4.1E+05
2.6E+05
2.5E+05
4.1E+05
2.2F+05
100
48
9.3E+05
4,3E+05
8.3E+05
4.2F+05
4.0E+05
8.3E+05
3.4E+05
200
33
1.9E+06
6,4E+05
1.4E+06
5,8E+05
5.7E+05
1,4E+06
5.1E+05
PINS
10770
256
30
56
3.2E+05
3,1E+05
3,2E+05
3.1E+05
3,0E+05
3.2E+05
2.7E+05
50
5,4++05
5.1E+05
5.2E+05
4.9E+05
4.8E+05
5,2E+05
3.8E+05
100
13
1.1E+06
9.2E+05
9.9E+05
8.3E+05
8.2E+05
9.9E+05
4.7E+05
200
20
2,2E+06
1.6E+06
2.0E+06
1.5E+0
1.5E+06
2.0E+06
6.4E+05
CPLFW
11652 256
30
37
3,5E+05
2.2E+05
2.3E+05
2.2E+05
2.1E+05
2.5E+05
2.0E+05
50
48
5.8D00
3.5E+05
3.7E+05
3.4E+05
3.3E+05
4.1E+05
3.2E+05
100
1.2E+06
6.7E+05
7.1E+05
6.4E+05
6.4E+0S
8,1E+05
6.2E+05
200
53
2.3B406
1.3E+06
1.3E+06
1.2E+06
1.2E+06
1,6E406
1.2E+06
EYaleB 16128 256
30
27
4,8E+05
1.6E+05
2.3E+05
1.4++05
1.1E+05
4,0E+05
7,4++04
50
31
8.1E+05
1.7E+05
4.1E+05
1.7E+05
1.4E+05
7,2E+05
1.0E+05
100
1.6E+06
2.7E+05
8.3E+05
2.4++05
2.1E+05
1.3E+06
1.7E+05
200
59
3.2E+06
3.3E+05 1.1E+06
3.0E+05
2.7E+05
1,9E+06
2.2F+05
Table 9: Per-iteration average of distance computation on large-scale datasets.
Datasets
d
k
Iter
Lloyd
Elkan
Ann
Exp
Ball-R
Ball-noR
Angle
L-CAS
30863 256
300
41
9.3E+06 8.2E+06
8.6E+06
8,0E+06
8.0E+06
8.9E+06
7.0E+06
500
29 1.5E+07 1.2E+07
1.3E+07
1.1E+07
1.1E+07
1.4E+07
8.9E+06
1000
24 3.40 1.2F+07
2.2F+07
1.1E+07
1.1E+07
2.3E+07
7.0E+06
L-CLBA 202599 256
5000
72 1.0E+09 4.2F+08
5.5E+08(1)
4,14)
4.1E+08(1)
4,1E+08(3)
5.6E+08(3)
2.8E+08
8000
39 1.6E+09 5.7F+08 8.5F+08(1)
5.4F+08(1)
5.4F+00(3)
8.6F+08(3)
3,0F+08
10000
40 2.0E+09 6.4E+08 1.0E+09(1) 6.0E+08(1) 6.0E+08(1) 1.0E+09(1) 3.0E+08
L-EDS 240000 784
500 137 1.2E+08 8.3E+07 1.2E+08(1) 8.7E+07(1) 8.7E+07(1) 1.2E+08(I) 5.8E+07
800 227 1.9E+08 1.2E+08 1.9E+08(1) 1.3E+08(1) 1.3E+08(1) 1.9E+08(1) 8,0E+07
1000 144 2.4E+08 1,4E+08 2.3E+08(1) 1.5E+08(1) 1.5E+08(1) 2,4E+08(1) 9.3E+07
L-YTF 621126 256
500 246 1E+08 1.7E+08 2.5E+08(1) 1.7E+08(1) 1.7E+08(5) 2.6E+08(5) 1.4E+08
1000 136 6,2E+08 2.5E+08 4.6E+08(1) 2.5E+08(1) 2.5E+08(5) 4.7E+08(5) 1.9E+08
2000 100 1.2E+09 3.1 8.4E+08(1) 3.0E+08(1) 3.0E+08(1) 8. 8.7E+08(1) 1.9E+08
The numbers in parentheses indicate the iteration counts recorded for the corresponding algorithms.
16

---

Published as a conference paper at ICLR 2026
Table 10: Per-iteration average running time (s) on middle-scale datasets
Dataset
d
k
Iter
Lloyd
Elkan
Ann
Exp
Ball-R
Ball-noR
Angle
Digits
4000
256
30
32
3.6F-04
3.5E-04
1.5E-02
2.1F-02
4.0E-03
4.0F-03
3.2E-04
50
22
4.6F-04
4.3F-04
2.5E-02
3.2E-02
6.4F-03
5.1E-03
4.2E-04
100
7.6B-04
7.1E-04
5.0E-02
5.9E-02
1.3E-02
9,4E-03
5.8E-04
200
19
1.6E-03
1.2E-03
1.0E-01
9.9E-02
2.9E-02
2.0E-02
1.0E-03
Corel5k
5000
423
30
39
6.5E-04
5.8E-04
3.2E-02
3.2E-02
8.9E-03
1.1E-02
5,4E-04
50
66
8.8E-04
7.6E-04
5.1E-02
4.25-02
1.26-02
1.2E-02
6.4E-04
100
1.5E-03
1.1E-03
1.0E-01
6.7E-02
2.5E-02
2.0E-02
8.3E-04
200
27
2.8E-03
1.76-03
1.9E-01
1.0E-01
4,55-02
4.3E-02
1.4E-03
CNBC
6574
256
30
44
5.7E-04
4.6E-04
2.20:02
2.0E-02
4.9E-03
8,4E-03
4.2E-04
50
35
7.6E-04
7.5E-04
3.3E-02
3.6E-02
6.9E-03
8.6E-03
7.2E--4
100
53
9.6E-04
9.7E-M
6.4E-02
5.6E-02
1.2E-02
1.3E-02
8.2E-04
200
23
2.0E-03
1.3E-03
1.2E-01
8.1E-02
2.5E-02
2.5B-02
1.1E-03
Coil100
7200
1024
30
45
2.1E-03
2.0E-03
1.25-11
1.3E-01
3.7E-02
4.3E-02
1.7E-03
50
39
2.9E.03
2.3E-03
2.3E-01
1.5E-01
4.1E-02
4.SE-02
2.1E-03
100
32
4.6F-03
3.2F-03
3.0E-01
2.1F-01
6.2F-02
6.3F-02
3.1E-03
200
29
8.1E-03
4.9F-03
4.9E-01
2.7E-01
1.18-01
1.1E-01
4.7E-03
Isolet
7797
617
30
1.7E-03
1.3E-03
8.1E-02
8.4E-02
2,2H-02
2.5E-02
1.2E-03
50
26
2.4E-03
1.6E-03
1.3E-01
1.2E-01
3.1E-02
2.9E-02
1.4E-03
100
39
2.9E-03
2.6E-03
2.7E-01
2.1E-01
5.8E-02
4.7E-02
2.3E-03
200
6.IE-03
5.0E-03
5.1E-01
3.6E-01
1.3E-01
9,4E-02
4.1E-03
RaFD
8040
256
30
8.4B-04
5.8E-04
1.7E-02
1.8E-02
4.3E-03
4.0E-03
4.6E-04
50
1,2E-03
7.3E-04
2.7E-02
2.6E-02
7.65-03
7.3E-03
6.2E-04
100
11
1.6E-03
9.6E-04
5.7E-02
3.8E-02
1,4B-02
1.1E-02
6.2E-04
200
26
2.
1.2E-03
1.E401
4.6E-02
2.18-02
2.2E-02
7.9E-04
USPS
9298
256
30
32
9.3E-04
6.7E-M
3.1E-02
3.0E-02
9.2F-03
9.5E-03
6.1E-04
50
9.8E-04
7.8E-04
5.1E-02
4. 4.1
1.3F-02
1.2F-02
6.4E-04
100
48
1,4F-03
1.1E-03
1.0F-01
6.2F-02
2.1F-02
1.8F-02
1.0E-03
200
33
2.5E-03
1.7E-03
1.9E-01
9.0E-02
3.5E-02
3.2E-02
1.5E-03
PINS
10770
256
30
56
8.3E-04
8.3E-04
3.6E-02
4.0E-02
8.9E-03
9.2E-03
6.5E-04
50
1.5E-03
1.1E-03
6.0E-02
6.3E-02
1.2E-02
1.1E-02
9.1E-04
100
2.25-03
1.8E-03
1.2E-01
1.1E-01
2.3E-02
1.8E-02
1.1E-03
200
20
2.9E-03
2.9E-03
2.6E-01
2.1E-01
5.3B-02
3.7E-02
1.6E-03
CPLFW
11652 256
30
37
8.9E-04
8.0E-04
3,5E-02
3,4E-02
8.7B-03
8.6E-03
7.9E-04
50
48
1,2E-03
9.1E-M
S.3E-02
4.9E-02
1.1E-02
1.1E-02
7.8E-04
100
1.8E-03
1.4E-03
1.0E-01
9.2E-02
1.6E-02
1.7E-02
1.2E-03
200
53
3.0E-03
2.5E-03
1.9E-01
1.8E-01
3.0E-02
3.0E-02
2.2E-03
EYaleB 16128 256
30
27
1.6E-03
8.2E-04
3.7E-02
3.3E-02
1.4E-02
1.6E-02
6.8E-04
50
31
2.0E-03
L.0
6.7E-02
4.0E-02
1.7E-02
2.0E-02
8.4E-04
100
2.7B-03
1.5E-03
1.3E-01
6.3E-02
2.0E-02
3.1E-02
1.2E-03
200
59
4.1E-03
1.8E-03
1.8E-01
7.7E-02
3.0F-02
4.1F-02
1.6E:03
Table 11: Per-iteration average running time (s) on large-scale datasets
Datasets
d
k
Iter
Lloyd
Elkan
Ann
Exp
Ball-R
Ball-noR
Angle
L-CAS 30863 256
300
41
1.1B-02 1.3E-02
1.1E+00
1.1E+00
1.8E-01
1.5E-01
1.0E-02
500
1.7B-02
1.9E-02
1.8E+00
1.7E+00
3.3E-01
2.6B-01
1.6E-02
1000
3.4E-02
2.3F-02
3.3E+00
1.8E+00
6.7F-01
6.0F-01
1.7F.02
L-CLBA 202599 256
5000
72
7.9E-01
7.2F-01
7.99-01(1)
1,36400(1)
1.77400))
(37
5.3E-00
8000
39 1.3E+00
1.1=+00
(.524402())
6,15-40103339944110)
7,3E-01
10000
40 1.6E+00 1.4E+00
1.7E+02(1)4.8E4022/ 4.2E+01(1)5.4E40000
(1)5--01))..E-0111 8.5E-01
L-EDS 240000 784
500
3.8E-01 3.1E-01
7.SE=-0(())..6E-01)) 1.4E+01(I) (2E+01(1)2.44444
800
227 4.0E-01 4.5E-01
1.4E+02(1) (.77-+0221)2.11-00000
1.SE+01(I) 3.4E-01
1000 144 5.3E-01 5.5E-01
2.2E+01(1) 1.7E+01(1)
(1)22-01) 4,2E-01
L-YTF 621126 256
500 246 2.55-01 2.6E-01
3.2E+01(1) 3.8E+00X5) 4,155)411E+44 4.10+00(5) 2.2E-01
1000 136 4.7B-01 3.9E-01 4.33-00))) 6.0E+00K5) 6.8E+00(5) 3.2E-01
2000 100 9.5E-01 5.2E-01 1.1E+02(1) 9,4E+01(1) 9.1E+00(1) 1.2E+01(1) 3.6E-01
The numbers parentheses indicate iteration counts recorded for the corresponding algorithms.
17

---

Published as a conference paper at ICLR 2026
A.4 THE NUMBER OF CALCULATED DISTANCES IN EACH (RESUTI ON ALL DATASETS)
In Figure
and
we only show part of the results. Here are the full results.
8.0e+05
1.0e+06
9.0e+-05
number distances
number of distannce
7.0e+05
8.0++05
live
7.0e+055
6.0e+ 05
6.0e+05
5,0e+
5.0e+05
4.0e+05
3.0e+05
5
10
15
5
10
15
20
25
Iteration
Iteration
(a) Digits
(b) Corel5k
1.4e+06
1.2e+06
number distances
number of distances
1.0e+006
1.0e+066
0.0e+05
8,0e+05
6.0e+05
6.0e+05
4.0e+05
4.0e+05
Iteration
Iteration
(c) Coil100
(d) CNBC
1.5e+06
1.4e+06
number of distances
1.2e+06
1.0e++06 06
1.2e+06 06
7.5e+05
1.e+06
5.0e+05
2.5e+05
8-
8.0e+05
2
10
15
10
155
20
Iteration
Iteration
(c) Isalet
(f) RaFD
Figure The number of calculated distances each iteration on datasets. (Part
18

---

Published as a conference paper at ICLR 2026
2.0e+06
number of distances
1.80+ 06
0.2e+00
1.5e+ 06
of
1.0e+
1.2e+006
05
1.0e+06
5.0e+ 05
7.5e+05
10
15
25
15
Iterationn
Iteration
(a) USPS
(b) PINS
0.2e+06
3.0e+06
number distances
number of distances
2.5e+06
2.0e+06
Lue eage
2.0e+06
1.8e+06
1.5e+06
1.6e+06
1.0e+06
1.4e+06
5.0e+05
10
20
30
40
50
10
20
40
50
Iteration
Itaration
(c) CPLFW
(d) EYaleB
3.0e+07
2.0e+09
1.8e+09
Ball noBi has some number distance computaaton
2.5
Rumber of distances
1.5e+09
2.0e+07
1.2e+09
Ball R has the sartie computatiion
of
1.0e+09
1.5e+07
7.5e+08
1.0e+07-
5.0e+08
5.0e+06
2.5e+08
10
15
20
Iteration
L-CAS
(f) L.CLBB
1.2e+09
2.2e+08
number distances
1.0e+09
number of distances
2.0e+08
ball soli
8.0e+08
Ball Ball-RWa RR and Expo 9 number of distance computation
1.Be+08
Fuitieer of distance
6.0e+08
ligs
1.5e+08
4.0e+08
1.2e+08
2.0e+08
1.0e+08
20
60
80
100
25
100
125
150
Iteration
Iteration
(g) L-YTF
(h) L-EDS
Figure 6: The number of calculated distances in each iteration all datasets. (Part (Due the large data volume and long computation time, only the number of distance calculations in the first iteration was recorded for Ann, Exp. Ball-R, and Ball-noR on the L-CLBA, L-YTF, and L-EDS datasets.)
19


Reviews:
[{"review_id": "MnHOmb2jlE", "reviewer": "Reviewer_5Q16", "summary": "The paper proposes an angle-based acceleration approach for the K-means clustering algorithm, referred to as Angle-means. The main idea is to use angular bounds to prune unlikely candidate centers in the assignment step, thereby reducing the number of distance computations. The authors derive a theoretical bound (Theorem 2) based on geometric analysis and compare Angle-means with several existing acceleration variants, including Annulus, Exp-ns, and Ball K-means. Experimental results show that Angle-means achieves comparable clustering accuracy but faster runtime than baseline methods on multiple datasets.", "questions": "1. The paper evaluates five algorithms: Lloyd, Annulus, Exp-ns, Ball K-means, and Angle-means. The latter four are intended as accelerated versions of Lloyd. According to Table 3, all of them reduce the number of distance computations compared to Lloyd. However, in Table 4 (runtime), only Angle-means runs faster, while Annulus, Exp-ns, and Ball K-means are actually slower. Could the authors clarify why this happens? Is it mainly due to additional preprocessing or bookkeeping costs?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Solid geometric reasoning behind the angular pruning criterion. \n\n2. Simple algorithmic design that is easy to implement and integrate with existing K-means frameworks. \n\n3. Empirical results demonstrate consistent, though moderate, runtime gains compared to several baseline acceleration methods. \n\n4. Theoretical guarantees (Lemma 1, Theorem 2) are sound and clearly derived.", "weaknesses": "1. The proposed algorithm is an acceleration method for exact K-means, but lacks comparison with approximate K-means algorithms (e.g., uniform sampling, coreset-based methods) in terms of runtime, clustering cost, and accuracy (ACC). Such comparisons would help clarify the trade-off between precision and efficiency.\n\n2. While the geometric intuition is sound, the theoretical analysis remains relatively elementary. A more rigorous convergence analysis or worst-case guarantee (beyond per-iteration complexity) would significantly strengthen the contribution. In particular, it would be useful to discuss whether the parameter p in the time complexity can be expressed or bounded in expectation as a function of 𝑘.\n\n3. When the number of clusters 𝑘 is very large, the proposed acceleration scheme seems to offer limited benefit, as the overhead of managing angular relationships may offset the gain from pruning.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vaSs3rX489", "reviewer": "Reviewer_NsYa", "summary": "The paper introduces Angle k-means, an exact and parameter-free acceleration of standard k-means that exploits geometric angle relations between cluster centers to skip redundant distance calculations. By precomputing center distances and angles, it preserves identical results to Lloyd’s k-means while greatly improving efficiency. Compared with Ball k-means and Exponion k-means, it achieves faster clustering without accuracy loss. Theoretical analysis shows linear complexity in data size, and experiments on 14 datasets report up to 90% fewer distance computations.", "questions": "See weakness", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The algorithm relies on clear geometric intuition (angle constraints) without complex modifications.\n\nAchieves up to 80–90% reduction in distance computations on real datasets.\n\nWell-suited for low- and mid-dimensional large-scale datasets in industrial use.", "weaknesses": "The idea is largely derived from existing geometric pruning strategies, with only minor reformulation via angular constraints.\n\nThe approach strongly resembles recent TILB k-means methods that also exploit triangle-inequality-based geometric filtering, yet the paper provides no comparison or discussion.\n\nThe algorithm only applies to Euclidean metrics and cannot extend to cosine or Mahalanobis distances.\n\nAngle-based pruning loses effectiveness in high-dimensional spaces.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mS8uLc5u6Q", "reviewer": "Reviewer_qQbA", "summary": "This paper presents a simple improvement to Elkan’s triangle inequality pruning method, while avoiding the need to store O(nk) distances, using the law of cosines.They give a theoretical motivation and provide experiments, comparing the average number of distance computations per iteration and the average time per iteration to other hyper-parameter free exact k-means algorithms.", "questions": "Please clarify the discrepancy between the running time of the pseudo code and the definition of p. \n\nPlease provide an explanation for why Lemma 1, Theorem 2 and the case distinction are present.\n\nIn the abstract, it is stated that the method maintains linear time complexity wrt dimensionality, but the running times given in Section 3 are independent of dimension. Why?\n\nWhat does “Setup” refer to in Table 1?\n\nAre lines 11-13 of Algorithm 1 necessary?\n\nDo any of the parameterised methods have sensible default parameters? It seems strange to ignore them in the experiments.\n\nCan you explain the discrepancy between the difference in the average number of distance computations and the average running times (Table 3 vs Table 4). On some datasets (eg. Isolet, k=30), your method uses slightly fewer distance calculations, but runs over 10x faster.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 1, "strengths": "The running times of the experiments look very promising.", "weaknesses": "Looking at the pseudo code of Algorithm 1, the loop on line 15 only stops early when the triangle inequality bound is reached (the current center is at least twice the distance from the existing one), regardless of the angle criteria. Thus the p in the running time does not refer to the average number of candidate centers per sample according to the definition of B, but rather to the average number of candidate centers under the triangle inequality bound. This gives a theoretical running time no better than what the standard triangle inequality method would give. It is not obvious how to remedy this since the angle criteria is not monotonic over centers and so cannot be used to stop early on its own. Dealing with this issue would be a significant result.\nThe presentation of the paper severely obfuscates the underlying idea. Lemma 1, Theorem 2, and the case distinction on page 4 appear to be completely unnecessary since equation (10) can be derived directly from the law of cosines and  (10) is the only statement required to derive the pruning rule given in the caption for Figure 1 c) that is used in Algorithm 1. Deriving (10) from the law of cosines would make the argument far simpler and easier to follow.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "TOu27Eqi9e", "reviewer": "Reviewer_joh2", "summary": "This paper provides a new way to make exact k-means faster. Consider a point, its current center, and an alternative center. The basic idea is that if the angle between the point and the alternative is large enough, and the alternative is far away from the current center, then it cannot be the a better center for the point.", "questions": "Does Figure 1(c) show that the new method is a guaranteed improvement over Elkan's method (in number of distance computations, not necessarily in clock time)? Which other methods have a similar guarantee?\n\nTables 3, 4, 5, 6 seem to show that speedups never exceed 10x, even with thousands of centers. What is the fundamental reason for this lack of great speedup?\n\nIn high dimensions, any two random points are almost certainly almost orthogonal, so their angle is pi/2. Is this fact relevant to the algorithm of this paper? What does it imply for how often the condition will be true in line 19 of the algorithm?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "This paper presents a new technique for making k-means faster. The idea is explained carefully and the experimental results are favorable. \n\nThe general idea is to calculate a score for each pair of centers, and then to use these scores to recognize for each point that many alternative centers are more distant than the point's current center. This idea (lines 175 to 177) dates back to Elkan (2003) and is illustrated in Figure 1(a) of this submission. Figure 1(c) shows that the new method can eliminate additional centers.\n\nThe new idea is a significant increment over previous research.", "weaknesses": "Section 2 should not just list and summarize previous methods. It should say carefully how the methods compare in time and space complexity, and in empirical speed.\n\nThere are numerous small imperfections in the English of the submission, such as a comma splice at the end of Equation 5.\n\nWhat is the point of Figure 1(b)? Why not just have 1(c)? Only 1(c) is used in the actual algorithm.\n\nBesides Lloyd's, why were Annulus, Exp-ns, and Ball hosen for comparison and not other previous algorithms? In particular, including Elkan's method would be interesting since Figure 1(c) seems to show that Angle k-means is strictly better.\n\nTables 3, 4, 5, 6 would be more clear if they showed other methods as a percentage of Lloyd's as a baseline.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)