# Bound for Expectation of Singular Value

In my case, $$X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$$ is a function of Rademacher variables $$\boldsymbol{\delta}\in\{1,-1\}^M$$ with $$\delta_i$$ independent uniform random variables taking values in $$\{−1, +1\}$$. $$X_{\boldsymbol{\delta}}=[\sum_{i=1}^{I_1}\delta_{i}\mathbf{x}_{i},\sum_{i=I_1+1}^{I_2}\delta_{i}\mathbf{x}_{i},...,\sum_{i=I_{M-1}+1}^{I_M}\delta_i\mathbf{x}_{i}]$$ is a group-wise sum with known $$I_1,I_2,...,I_M$$ and non-singular $$X=(\mathbf{x}_1,\mathbf{x}_2,...,\mathbf{x}_N)\in\mathbb{R}^{d\times N}$$ where $$N>M\gg d$$.

Given that $$\sigma_i(X_{\boldsymbol{\delta}})$$ denotes $$i$$-th smallest singular value, how can I find the lower bound of the expectation $$\underset{\boldsymbol{\delta}}E\left[\sum_{i=1}^{k} \sigma_{i}^{2}\left(X_{\boldsymbol{\delta}}\right)\right]$$ assuming $$k?

Note: I can find an upper bound by Jensen’s inequality and concavity of sum of $$k$$ smallest eigenvalue, but I am curious about whether it is possible to get a lower bound.

I have also posted the question here.