Let $\mathbf{A}\in\mathbb{R}^{k\times n}$ and $\mathbf{B}\in\mathbb{R}^{d\times n}$ be independent matrices with i.i.d. $\mathcal{N}(0,1)$ entries. I'm interested in lower bounding the minimum eigenvalue of the Khatri-rao product of these two matrices with high probability. The Khatri-rao product of two matrices $\mathbf{A}$ (with columns $\mathbf{a}_1, \mathbf{a}_2, \ldots,\mathbf{a}_n$) and $\mathbf{B}$ (with columns $\mathbf{b}_1, \mathbf{b}_2, \ldots,\mathbf{b}_n$) denoted by $\mathbf{A}\odot \mathbf{B}$ is the $(kd) \times n$ matrix whose columns consists of the Kronecker product of the columns of $\mathbf{A}$ and $\mathbf{B}$. That is the i-th column of $\mathbf{A}\odot \mathbf{B}$ is equal to $\mathbf{a}_i\otimes \mathbf{b}_i\in\mathbf{R}^{(kd)\times 1}$. I'm interested in lower bounds on $\sigma_{min}(\mathbf{A}\odot \mathbf{B})$ that hold with high probability. I'm interested in the regime where $kd=c n$ with $c$ a fixed numerical constant smaller than $1$ say $0.05\le c \le 0.5$.

Let $\mathbf{A}\in\mathbb{R}^{k\times n}$ and $\mathbf{B}\in\mathbb{R}^{d\times n}$ be independent matrices with i.i.d. $\mathcal{N}(0,1)$ entries. I'm interested in lower bounding the minimum eigenvalue of the Khatri-rao product of these two matrices with high probability. The Khatri-rao product of two matrices $\mathbf{A}$ (with columns $\mathbf{a}_1, \mathbf{a}_2, \ldots,\mathbf{a}_n$) and $\mathbf{B}$ (with columns $\mathbf{b}_1, \mathbf{b}_2, \ldots,\mathbf{b}_n$) denoted by $\mathbf{A}\odot \mathbf{B}$ is the $(kd) \times n$ matrix whose columns consists of the Kronecker product of the columns of $\mathbf{A}$ and $\mathbf{B}$. That is the i-th column of $\mathbf{A}\odot \mathbf{B}$ is equal to $\mathbf{a}_i\otimes \mathbf{b}_i\in\mathbf{R}^{(kd)\times 1}$. I'm interested in lower bounds on $\sigma_{min}(\mathbf{A}\odot \mathbf{B})$ that hold with high probability. I'm interested in the regime where $kd=c n$ with $c$ a fixed numerical constant larger than $1$ say $2\le c \le 20$.