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 ...
- 3,381
9
votes
Accepted
Almost uniform sampling implies approximate counting
Given a set $A \subseteq B$, if you can sample uniformly from $B$, then you can estimate by repeated sampling the probability of the event that the sample is in $A$. That is, you can approximate the ...
- 9,566
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 ...
- 23.7k
6
votes
Coreset and VC dimension
Samples provides you only with statistical guarantees. For a fixed $\epsilon$ whatever you compute would hold for everything "except" $(1-\epsilon)$ fraction (I am being very informal here). Thus, ...
- 9,566
5
votes
Sample complexity of distinguishing two Gaussian distributions?
I don't have a solution to this problem, but the analogous case where the two distributions are discrete has been analyzed in the cryptographic literature.
Suppose we want to distinguish between two ...
- 10.5k
4
votes
Accepted
Determining the number of clusters using property testing algorithm
First. One can do better as far as the sampling - at least if $d$ is large - $O(\frac{kd \log k}{\epsilon} \log \frac{1}{\epsilon})$ should follow easily from relative approximations http://sarielhp....
- 9,566
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 ...
- 14.8k
3
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 ...
- 2,743
3
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 ...
- 10.1k
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 ...
- 46
3
votes
Accepted
Approximating the value of k in $k$-mean clustering problem
This problem is known as Geometric Set Cover (which deals with covering with different shapes, so unit balls are a special case). I'm unaware of any relation to $k$-means which is a clustering ...
- 9,378
2
votes
What is the proper role of verification in quantum sampling, simulation, and extended-Church-Turing (E-C-T) testing?
The gist of the question is, given that quantum probability is a source of true randomness, how does that effect the extended (or efficient, or polynomial-time) Church-Turing thesis?
The answer is ...
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 ...
- 10.1k
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 ...
- 600
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 $...
- 4,341
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 ...
- 217
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 $m$ ...
1
vote
Evaluating the expected value of negatively correlated random variables
I think that in principle it would take exponential time to compute this probability exactly. Hence sampling would be the only option here, although you would have to settle for additive and ...
- 231
Only top scored, non community-wiki answers of a minimum length are eligible