Given a $(p,p)$ symmetric positive definite matrix $\mathbf{H}$ of rank $k\le p$, I am looking for a (possibly efficient) way to generate a set of $k$ vectors $\alpha_i\in\mathbb{R}^p$ uniformly distributed under the constraint $$\sum_{i=1}^k \alpha_i \alpha_i^\text{T} = \mathbf{H}$$

If needed I can further assume that there exists a set of $k$ vectors $\beta_i\in\mathbb{R}^p$ such that $\mathbf{H}$ is constructed as $$\sum_{i=1}^k \beta_i \beta_i^\text{T} = \mathbf{H}$$

Note that, despite the title, this is unrelated to simulating a Wishart in that $\mathbf{H}$ is fixed and the $\alpha_i$ are not $\text{N}_p(0,\mathbf{I}_p)$ variates.

Given a $(p,p)$ symmetric positive semi-definite matrix $\mathbf{H}$ of rank $k\le p$, I am looking for a (possibly efficient) way of generating a set of $k$ vectors $\alpha_i\in\mathbb{R}^p$ uniformly distributed under the constraint $$\sum_{i=1}^k \alpha_i \alpha_i^\text{T} = \mathbf{H}$$

If needed I can further assume that there exists a set of $k$ vectors $\beta_i\in\mathbb{R}^p$ such that $\mathbf{H}$ is constructed as $$\sum_{i=1}^k \beta_i \beta_i^\text{T} = \mathbf{H}$$

Note that, despite the title, this is unrelated to simulating a Wishart in that $\mathbf{H}$ is fixed and the $\alpha_i$ are not $\text{N}_p(0,\mathbf{I}_p)$ variates.

Given a $(p,p)$ symmetric positive definite matrix $\mathbf{H}$ of rank $k\le p$, I am looking for a (possibly efficient) way to generate a set of $k$ vectors $\alpha_i\in\mathbb{R}^p$ uniformly distributed under the constraint $$\sum_{i=1}^k \alpha_i \alpha_i^\text{T} = \mathbf{H}$$

If needed I can further assume that there exists a set of $k$ vectors $\beta_i\in\mathbb{R}^p$ such that $\mathbf{H}$ is constructed as $$\sum_{i=1}^k \beta_i \beta_i^\text{T} = \mathbf{H}$$

Note that, despite the title, this is unrelated to simulating a Wishart in that $\mathbf{H}$ is fixed and the $\alpha_i$ are not $\text{N}(0,1)$ variates.

Given a $(p,p)$ symmetric positive definite matrix $\mathbf{H}$ of rank $k\le p$, I am looking for a (possibly efficient) way to generate a set of $k$ vectors $\alpha_i\in\mathbb{R}^p$ uniformly distributed under the constraint $$\sum_{i=1}^k \alpha_i \alpha_i^\text{T} = \mathbf{H}$$

If needed I can further assume that there exists a set of $k$ vectors $\beta_i\in\mathbb{R}^p$ such that $\mathbf{H}$ is constructed as $$\sum_{i=1}^k \beta_i \beta_i^\text{T} = \mathbf{H}$$

Note that, despite the title, this is unrelated to simulating a Wishart in that $\mathbf{H}$ is fixed and the $\alpha_i$ are not $\text{N}_p(0,\mathbf{I}_p)$ variates.