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11 votes
Accepted

Sampling monotone Boolean functions

In [1], the paper where Propp & Wilson introduce the "coupling from the past" technique for MCMC sampling, they also outline a "heat bath algorithm" applicable to a certain kind of spin system ...
mhum's user avatar
  • 3,392
8 votes
Accepted

Qubit gates in google supremacy

Based on a fairly cursory inspection of their paper, IBM is clearly aware of the spatial locality. They seem to in fact have taken spatial locality into account when designing their simulation. They ...
Peter Shor 's user avatar
4 votes

Is uniform convergence faster for low-entropy distributions?

This is very much a partial answer to my question. I'm hoping for a much better bound (or a counterexample). I managed to show a very weak bound. It is not very useful, but it does at least show that ...
Thomas's user avatar
  • 2,803
4 votes

Is uniform convergence faster for low-entropy distributions?

First, let's use McDiarmid's inequality to conclude that $$\mathbb{P}\left[|| \bar X - \mu ||_\infty \ge \mathbb{E}|| \bar X - \mu ||_\infty + \varepsilon \right] \le e^{-2n\varepsilon^2},$$ so it ...
Aryeh's user avatar
  • 10.6k
4 votes
Accepted

How many samples are needed to reconstruct a path?

Here’s a simple idea: try a certain number $s$ of samples, and for each vertex, record how many times its colour agreed with the provided bit. For vertices outside the path, they agree with ...
Emil Jeřábek's user avatar
3 votes
Accepted

Coreset and VC dimension

The answer of Har-Peled is mainly regarding problems where you wish to cover points by shapes (e.g. balls). A strong relation to eps-nets and hitting sets can be found e.g. in his paper here For the ...
Dan Feldman's user avatar
2 votes

Is uniform convergence faster for low-entropy distributions?

We have largely resolved the question for product measures. I'm going to change the notation from the OP to be in line with our paper, https://arxiv.org/abs/2209.04054 I'll be writing $\mu$ rather ...
Aryeh's user avatar
  • 10.6k
2 votes
Accepted

Sample complexity for learning Boltzmann Distribution parameters

Under appropriate smoothness conditions (satisfied by the Boltzmann distribution), the maximum likelihood is asymptotically normal. Meaning: the vector $\sqrt n(\theta-\hat\theta)$ will have ...
Aryeh's user avatar
  • 10.6k
2 votes
Accepted

Lower bound for the OR problem

After some more searching, I managed to find a proof in these lecture notes [1]. The proof goes via Yao's principle and the lower bound is n/3. If someone knows of a published paper or a book that I ...
user2316602's user avatar
1 vote
Accepted

Sample complexity lower bound to learn the mode (the value with the highest probability) of a distribution with finite support

From the DKW inequality, it follows one can learn an arbitrary distribution over $\mathbb{R}$ (and in particular over $\{1,2,\dots,n\}$ to Kolmogorov distance* $\varepsilon$ with probability at least $...
Clement C.'s user avatar
  • 4,471
1 vote

Sample Complexity for Order Statistics

One way to approach this problem is via the CDF transformation. Consider $Z=F(X)$. We know $Z$ is uniformly distributed in $[0,1]$. Let $Z_{(1)},...,Z_{(m)}$ be the order statistics of these $m$ ...
Vaishakh Ravi's user avatar
1 vote
Accepted

How well do subspaces hit sets

Here's an $\Omega(\log(\frac{1}{\epsilon\cdot \delta}))$ lower bound following the discussion here with Yuval, and another one with Ross Berkowitz. I assume for simplicity that the sampling process ...
relG's user avatar
  • 209

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