Shashwat Agrawal

Shashwat Agrawal, Amitabha Bagchi, Rajendra Kumar

This paper extends the Kikuchi method to give algorithms for decisional $k$-sparse Learning With Errors (LWE) and $k$-sparse Learning Parity with Noise (LPN) problems for higher moduli $q$. We create a Kikuchi graph for a sparse LWE/LPN instance and use it to give two attacks for these problems. The first attack decides by computing the spectral norm of the adjacency matrix of the Kikuchi graph, which is a generalization of the attack for $q=2$ given by Wein et. al. (Journal of the ACM 2019). The second approach computes non-trivial closed walks of the graph, and then decides by computing a certain polynomial of edge labels in the walks. This is a generalization of the attack for $q=2$ given by Gupta et. al. (SODA 2026). Both the attacks yield new tradeoffs between sample complexity and time complexity of sparse LWE/LPN.

Poojan Shah, Shashwat Agrawal, Ragesh Jaiswal

The $k$-$\mathtt{means}$++ seeding algorithm (Arthur & Vassilvitskii, 2007) is widely used in practice for the $k$-means clustering problem where the goal is to cluster a dataset $\mathcal{X} \subset \mathbb{R} ^d$ into $k$ clusters. The popularity of this algorithm is due to its simplicity and provable guarantee of being $O(\log k)$ competitive with the optimal solution in expectation. However, its running time is $O(|\mathcal{X}|kd)$, making it expensive for large datasets. In this work, we present a simple and effective rejection sampling based approach for speeding up $k$-$\mathtt{means}$++. Our first method runs in time $\tilde{O}(\mathtt{nnz} (\mathcal{X}) + βk^2d)$ while still being $O(\log k )$ competitive in expectation. Here, $β$ is a parameter which is the ratio of the variance of the dataset to the optimal $k$-$\mathtt{means}$ cost in expectation and $\tilde{O}$ hides logarithmic factors in $k$ and $|\mathcal{X}|$. Our second method presents a new trade-off between computational cost and solution quality. It incurs an additional scale-invariant factor of $ k^{-Ω( m/β)} \operatorname{Var} (\mathcal{X})$ in addition to the $O(\log k)$ guarantee of $k$-$\mathtt{means}$++ improving upon a result of (Bachem et al, 2016a) who get an additional factor of $m^{-1}\operatorname{Var}(\mathcal{X})$ while still running in time $\tilde{O}(\mathtt{nnz}(\mathcal{X}) + mk^2d)$. We perform extensive empirical evaluations to validate our theoretical results and to show the effectiveness of our approach on real datasets.