寻求使$X^H X$对角元稀疏的矩阵X：求相关文献与软件指引

Great question—this is a classic group sparsity problem, a subset of sparse optimization that fits exactly what you're looking for. Let's break this down and give you concrete resources to tackle it.

First, let's clarify the math to simplify things: the diagonal entries of $X^H X$ are exactly the squared $\ell_2$-norms of $X$'s columns. That is:
$$(X^H X)_{ii} = |x_i|_2^2$$
So having a zero diagonal entry is equivalent to the $i$-th column of $X$ being entirely zero. Your goal is thus to find an $X$ (satisfying whatever other constraints you have—like linear equations from observations) that has as many zero columns as possible. This is exactly group-wise sparsity, where each "group" is an entire column of $X$.

Here are foundational and practical papers to build your understanding:

• Group LASSO (core framework): Yuan, M., & Lin, Y. (2006). Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(1), 49-67. This seminal work introduced group sparsity regularization, perfect for your column-wise sparsity goal.
• Reweighted Group LASSO (enhanced sparsity): Candès, E. J., Wakin, M. B., & Boyd, S. P. (2008). Enhancing sparsity by reweighted ℓ₁ minimization. Journal of Fourier Analysis and Applications, 14(5-6), 877-905. You can adapt the reweighting idea to group norms to encourage even more columns to drop to zero.
• Complex-Valued Case: Chi, Y., Chen, Y., & Gu, Y. (2013). Sparse recovery in complex-valued systems via reweighted ℓ₁ minimization. IEEE Transactions on Signal Processing, 61(19), 4684-4696. Since you're working with Hermitian transposes (complex matrices), this paper covers sparse recovery specifically for complex domains.

Depending on your preferred language, here are tools to implement this:

Python #

• CVXPY: The most flexible option for custom convex optimization. You can directly formulate your problem with group sparsity regularization. For example, if $X$ must satisfy $A X = B$ (linear constraints), here's a skeleton:

  import cvxpy as cp
  import numpy as np
  
  # Define problem dimensions and known complex matrices
  A = np.random.randn(10, 20) + 1j * np.random.randn(10, 20)
  B = np.random.randn(10, 5) + 1j * np.random.randn(10, 5)
  X = cp.Variable((20, 5), complex=True)
  
  # Group LASSO penalty: sum of ℓ₂ norms of each column (encourages zero columns)
  group_penalty = cp.sum(cp.norm(X[:, col], 2) for col in range(X.shape[1]))
  # Combine data fidelity + regularization (adjust λ to balance sparsity and fit)
  objective = cp.Minimize(cp.norm(A @ X - B, "fro") + 0.1 * group_penalty)
  constraints = []  # Add your specific constraints here (e.g., A@X == B)
  
  prob = cp.Problem(objective, constraints)
  prob.solve(solver=cp.SCS)  # SCS works well for complex-valued problems
  

• scikit-learn / pyglmnet: These libraries have pre-built group LASSO implementations for regression tasks. If your problem fits a regression framework (e.g., predicting $B$ from $A$ via $X$), these can save you time.
• PyTorch/TensorFlow: For large-scale or deep learning-integrated problems, you can add a custom group sparsity loss term (sum of column $\ell_2$ norms) to your training objective.

MATLAB #

• CVX Toolbox: Similar to CVXPY, you can formulate convex optimization problems with group sparsity using intuitive MATLAB syntax.
• lassoglm: Built-in function that supports group LASSO for generalized linear models, useful if your problem fits that structure.
• SPGL1: A specialized toolbox for sparse recovery that can be adapted to group sparsity scenarios.

Tune the regularization parameter $\lambda$ (the multiplier on the group penalty) to balance between fitting your constraints and maximizing the number of zero columns. Larger $\lambda$ will push more columns to zero, while smaller $\lambda$ prioritizes fitting your data/constraints.

内容的提问来源于stack exchange，提问作者Disenchanted Toad