# Maximum eigenvalue across subsamples

I have an $$N$$-dimensional vector of data, say $$X_{t}$$, with $$1 \leq t \leq T$$.

Of this vector $$X_{t}$$, I want to consider sub-vectors, say $$X_{t}^{b}$$, which are $$m$$-dimensional combinations of elements of the original vector $$X_{t}$$. In total, I have $${N}\choose{m}$$ of such combinations.

Note that $$N$$ can be a large number; I need to allow for $$N \rightarrow \infty$$. Also, I will choose $$m$$ in such a way that $$m$$ is "smaller" than $$N$$, e.g. $$m=O(N^{1/2})$$.

I want to compute the $$k$$-th eigenvalue ($$k$$ is user-defined) of the second moment matrix of each of these $$X_{t}^{b}$$s, i.e.

$$\lambda_{k}^{b} \left( \frac{1}{T} \sum_{t=1}^{T} X_{t}^{b} (X_{t}^{b})' \right).$$

Then, finally, I want to compute

$$\max_{1 \leq b \leq B} \lambda_{k}^{b},$$

where $$B$$ is defined as $${N}\choose{m}$$. In principle, I may need to compute also other measures such as the average of $$\lambda_{k}^{b}$$.

I have two questions:

1. Is this problem NP-hard? Is there a reference to back up either statement (i.e. "it is" or "it is not")?
2. Is there some heuristic to make the problem less computationally burdensome? Is there any way to prove, for a given heuristic, that the solution found by the heuristic is "very close" to $$\max\limits_{1 \leq b \leq B} \lambda_{k}^{b}$$ - e.g. by showing that the solution found by the heuristic and $$\max\limits_{1 \leq b \leq B} \lambda_{k}^{b}$$ are equal almost surely, or something similar?