SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Finally, we discuss our improved lower bounds for various spectral sum estimation problems. Analogous to Lemma 6, in the following lemma we show that to determine whether a graph has a triangle (i.e.,  $\operatorname{tr}\left(\mathbf{A}_G^3\right) > 0$ ) with constant probability, it suffices to determine whether  $\operatorname{tr}\left(\overline{\mathbf{A}}_G^3\right) > 0$ .
Lemma 8. If  $\operatorname{tr}(\mathbf{A}_G^3) = 0$ , then  $\operatorname{tr}(\bar{\mathbf{A}}_G^3) = 0$ , and if  $\operatorname{tr}(\mathbf{A}_G^3) > 0$  then  $\mathbb{P}\left[\left|\operatorname{tr}(\bar{\mathbf{A}}_G^3)\right| > 0\right] \geqslant 1/4$ .
The central idea of the proof is that if a triangle  $\{a,b\}, \{b,c\}, \{c,a\}$  exists in  $G$ , then amongst the 4 possible configurations of the Rademacher random signing variables  $\xi_{a,b}$  and  $\xi_{b,c}$ , at least one configuration must result in  $|\operatorname{tr}(\bar{\mathbf{A}}_G^3)| > 0$ . By following the proof of Theorem 15 from [33], and replacing their use of  $\mathbf{A}_G$  with a symmetric random signing  $\bar{\mathbf{A}}_G$ , we obtain an improved randomized version of their result by leveraging the smaller spectral radius of  $\bar{\mathbf{A}}_G$ . Theorem 5 follows by applying this result to the functions  $f$  that define the corresponding spectral sums (see supplementary material).
In this paper we provided improved upper and lower bounds on the problem of estimating and sketching effective resistances on expanders. On the algorithmic side we show how sketches tailored to  $\ell_1$  when carefully applied to asymmetric formulations of the quadratic form of the Laplacian pseudoinverse gave our results. On the lower bound side, we provided an alternative to the trace estimation approach of [33] for showing lower bounds and coupled it with techniques of randomly signing edges of the graph to obtain our results. Further, we showed that these techniques had broader implications for addressing algorithmic challenges in numerical linear algebra.
Beyond the natural open problem of improving both our upper and lower bounds towards bringing them together, there are interesting open problems in broadening the applicability of both our upper and lower bounds. For example, obtaining an  $\widetilde{O}(m\epsilon^{-1})$  time algorithm for estimating the effective resistance of all edges in a general (non-expander) graph and extending our  $\widetilde{\Omega}(n^2\epsilon^{-1/2})$  lower bounds to deterministic algorithms remain interesting open problems. We hope that the results of this paper provide useful tools for addressing each.
We thank Hongyue Li for helpful discussions and work on this project at various stages. Aaron Sidford was supported in part by a Microsoft Research Faculty Fellowship, NSF CAREER Award CCF-1844855, NSF Grant CCF-1955039, a PayPal research award, and a Sloan Research Fellowship. Ishani Karmarkar was supported in part by an NSF CAREER Award CCF-1844855, NSF Grant
CCF-1955039, a PayPal research grant, and a Stanford Institute for Computational and Mathematical Engineering (ICME) Fellowship. Rajat Vadiraj Dwaraknath was supported by a Stanford ICME Fellowship.
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