arXiv subjects

Dongrun Cai

Publications and source records attributed to Dongrun Cai.

5 recordsLinked to original sources

arXiv2026

Learning Multiband Signals and Fourier-sparse Signals

We consider efficient algorithms to learn multiband signals and Fourier-sparse signals. A mutliband signal has a Fourier transform supported by a bounded number of intervals, say $I_1 \cup I_2 \cdots \cup I_n$. There is a long line of research on multiband signals. In particular, Avron et al. showed an efficient reconstructing algorithm whose sample complexity is almost optimal. However, all previous algorithms for multiband signals consider the reconstructing problem in which the locations of $I_1,\ldots,I_n$ are given as a priori knowledge. On the other hand, although the problem of learning Fourier-sparse signals with $k$ arbitrary frequencies dates at least to Prony in 1795, designing efficient and robust learning algorithms is still an open problem. The state-of-the-art is an efficient algorithm of $\tilde{O}(k^3)$ samples and $\tilde{O}(k^{3 \omega})$ time from the very recent work by Cai et al., while the statistical upper bound is $\tilde{O}(k^2)$ samples. Let $[-1,1]$ be the time window in which the noise is $\ell_2$ bounded. 1. We show an efficient algorithm to recover the locations of the bands $I_1,\ldots,I_n$ in $\hat{x}$ within $\tilde{O}(n+\sum_i |I_i|)$ samples and $\tilde{O}(n+\sum_i |I_i|)$ time. Furthermore, combining this with the reconstructing algorithm by Avron et al. provides an efficient interpolation algorithm within $\tilde{O}(n+\sum_i |I_i|)$ samples. 2. We show that every $k$-Fourier-sparse signal $x$ admits a multiband approximation $z$ whose Fourier transform is of support size $|\mathrm{supp}(\hat{z})|=\tilde{O}(k^2)$. Furthermore, we show an interpolation algorithm for $k$-Fourier-sparse signals with $\tilde{O}(k^2)$ samples and $\tilde{O}(k^5)$ time.

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arXiv2026

Improved Algorithms for Learning Fourier-sparse Signals

A classical problem in sparse Fourier transforms, which dates back to the work by Prony in 1795 at least, is to learn a $k$-Fourier-sparse signal $x(t):=\sum_{j=1}^k \alpha_j e^{2 \pi \mathbf{i} f_j t}$ with arbitrary frequencies $f_1,\ldots,f_k$. We study this problem of learning $x(t)$ in a fixed time window $[-T,T]$ under adversarial noise with bounded $\ell_2$ norm, where the frequencies $f_1,\ldots,f_k$ may be "off-grid" -- arbitrarily located in a given bandlimit $[-F,F]$. In particular, our goal is to output a sparse interpolation $\tilde{x}$ such that $\tilde{x}(t) \approx x(t)$ in the time window $[-T,T]$. 1. Our first result shows that the sample complexity of interpolation is $k^2 \cdot O(\log \frac{k FT}{\epsilon})^2$. While its running time is $(\frac{k FT}{\epsilon})^{O(k)}$, this improves the previous upper bound $k^{4} \cdot (\log FT)^{O(1)}$ on the sample complexity substantially and leaves a gap of about $k$ to the lower bound $\Omega(k \log FT)$. 2. Our second result provides efficient algorithms to interpolate $x(t)$. The first algorithm takes $m=k^{3.75} \cdot (\log FT)^{O(1)}$ samples and $m^{\omega+o(1)}$ time ($\omega$ is the matrix multiplication exponent). Assuming that the growth of any $k$-Fourier-sparse signal cannot be significantly larger than the growth of the degree-$(k-1)$ Chebyshev polynomial -- specifically, $x(t) \le e^{k \cdot O\big( \sqrt{\frac{|t|}{T}-1} \big)} \cdot \underset{s \in [-1,1]}{\max} |x(s)|$ for any $t \notin [-T,T]$, the second algorithm further improves the sample complexity to $m'=k^{3} \cdot (\log FT)^{O(1)}$ and the time complexity to $(m')^{\omega+o(1)}$.

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arXiv2026

Sparsify Submodular Functions under Cardinality Constraints

Submodular sparsification generalizes the classical sparsification problems of graphs and matrices to summations of submodular functions. Given the summation $F(S):=f_1(S)+\cdots+f_m(S)$ of $m$ submodular functions $f_1,\ldots,f_m:\{0,1\}^n \to \mathbb{R}_{\ge 0}$. An size-$s$ sparsification of $F$ is a weight vector $w \in \mathbb{R}^m_{\ge 0}$ such that $w_1 f_1(S) + \cdots w_m f_m(S) \approx F(S)$ for every subset $S \subset [n]$. Motivated by the wide applications of submodular functions in data mining and economics, submodular sparsification has been studied in the last few years. For general submodular functions, Kenneth and Krauthgamer provided an efficient construction of size $O(n^3)$. Although several families of submodular functions admit sparsifiers of size $\tilde{O}(n)$, there is a lower bound $\Omega(n^2)$ on the size of sparsifiers by Cohen et al. In this work, we study whether cardinality constraints, such as restricting $S$ to subsets of size at most $k$, could reduce the size of sparsifiers or not. Namely, if the guaranty is $w_1 f_1(S) + \cdots w_m f_m(S) \approx F(S)$ for every $S$ in $[n]$ of cardinality at most $k$, are there sparsifiers of size smaller than $o(n^2)$? Our main result shows an efficient construction of size-$O(n k^2 \log n)$ sparsifiers for summations of arbitrary submodular functions. This improves the $\Omega(n^2)$ bound for the general setting. Then we consider the existence of size-$(k \log n)^{O(1)}$ sparsifiers under the constraint of cardinality at most $k$ and show several natural families do not admit such a small sparsifier. Technically, our algorithm applies the Lov\'{a}sz extension and Edmonds' greedy algorithm to extend Kenneth and Krauthgamer's approach. In particular, we provide an efficient algorithm to provide a tight estimate (up to a constant) of the sensitivity of each $f_i$ under cardinality constraints.

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arXiv2024

Revisit the Partial Coloring Method: Prefix Spencer and Sampling

As the most powerful tool in discrepancy theory, the partial coloring method has wide applications in many problems including the Beck-Fiala problem and Spencer's celebrated result. Currently, there are two major algorithmic methods for the partial coloring method: the first approach uses linear algebraic tools; and the second is called Gaussian measure algorithm. We explore the advantages of these two methods and show the following results for them separately. 1. Spencer conjectured that the prefix discrepancy of any $\mathbf{A} \in \{0,1\}^{m \times n}$ is $O(\sqrt{m})$. We show how to find a partial coloring with prefix discrepancy $O(\sqrt{m})$ and $\Omega(n)$ entries in $\{ \pm 1\}$ efficiently. To the best of our knowledge, this provides the first partial coloring whose prefix discrepancy is almost optimal. However, unlike the classical discrepancy problem, there is no reduction on the number of variables $n$ for the prefix problem. By recursively applying partial coloring, we obtain a full coloring with prefix discrepancy $O(\sqrt{m} \cdot \log \frac{O(n)}{m})$. Prior to this work, the best bounds of the prefix Spencer conjecture for arbitrarily large $n$ were $2m$ and $O(\sqrt{m \log n})$. 2. Our second result extends the first linear algebraic approach to a sampling algorithm in Spencer's classical setting. On the first hand, Spencer proved that there are $1.99^m$ good colorings with discrepancy $O(\sqrt{m})$. Hence a natural question is to design efficient random sampling algorithms in Spencer's setting. On the other hand, some applications of discrepancy theory, prefer a random solution instead of a fixed one. Our second result is an efficient sampling algorithm whose random output has min-entropy $\Omega(n)$ and discrepancy $O(\sqrt{m})$. Moreover, our technique extends the linear algebraic framework by incorporating leverage scores of randomized matrix algorithms.

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arXiv2023

Effective Resistances in Non-Expander Graphs

Effective resistances are ubiquitous in graph algorithms and network analysis. In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair $s$ and $t$. We consider the classical adjacency list model for local algorithms. While recent works have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of $n$ vertices and $m$ edges: 1.It needs $Ω(n)$ queries to obtain $1.01$-approximations of the effective resistance of an adjacent pair $s$ and $t$, even for graphs of degree at most 3 except $s$ and $t$. 2.For graphs of degree at most $d$ and any parameter $\ell$, it needs $Ω(m/\ell)$ queries to obtain $c \cdot \min\{d, \ell\}$-approximations where $c>0$ is a universal constant. Moreover, we supplement the first lower bound by providing a sublinear time $(1+ε)$-approximation algorithm for graphs of degree 2 except the pair $s$ and $t$. One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest.

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