Back to Search
Start Over
Sparse Convolution for Approximate Sparse Instance
- Publication Year :
- 2023
-
Abstract
- Computing the convolution $A \star B$ of two vectors of dimension $n$ is one of the most important computational primitives in many fields. For the non-negative convolution scenario, the classical solution is to leverage the Fast Fourier Transform whose time complexity is $O(n \log n)$. However, the vectors $A$ and $B$ could be very sparse and we can exploit such property to accelerate the computation to obtain the result. In this paper, we show that when $\|A \star B\|_{\geq c_1} = k$ and $\|A \star B\|_{\leq c_2} = n-k$ holds, we can approximately recover the all index in $\mathrm{supp}_{\geq c_1}(A \star B)$ with point-wise error of $o(1)$ in $O(k \log (n) \log(k)\log(k/\delta))$ time. We further show that we can iteratively correct the error and recover all index in $\mathrm{supp}_{\geq c_1}(A \star B)$ correctly in $O(k \log(n) \log^2(k) (\log(1/\delta) + \log\log(k)))$ time.
- Subjects :
- Computer Science - Data Structures and Algorithms
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2306.02381
- Document Type :
- Working Paper