md_files/科研/草稿.md

Here's the reformatted version with Markdown + LaTeX, including explanations for the * symbol:

---

Input:

• Ordered singular value estimates \tilde\sigma_1 \ge \tilde\sigma_2 \ge \cdots \ge \tilde\sigma_r,
• Budget s \ge 0.

---

1. Prefix Sum Array Initialization

S = [0] * (r+1)

Note

: The * symbol here is Python's list repetition operator, used to create a length (r+1) array filled with zeros. In formal papers, this would typically be written as:

S_0 = S_1 = \cdots = S_r = 0

or more concisely:

S = \mathbf{0}_{r+1}.

2. Absolute Value Prefix Sum Computation

For \kappa=1,2,\dots,r:

S[\kappa] = S[\kappa-1] + \bigl|\tilde\sigma_\kappa\bigr|.

3. Search Threshold Calculation

\theta = S[r] - s.

4. Binary Search on S[0\ldots r]

Find the minimal \kappa satisfying:

S[\kappa] \ge \theta.

Pseudocode (Python-style):

low, high = 0, r
while low < high:
    mid = (low + high) // 2
    if S[mid] < θ:
        low = mid + 1
    else:
        high = mid
κ = low

---

Key modifications:

1. Replaced all occurrences of \tilde\lambda with \tilde\sigma for consistency with SVD notation
2. Maintained proper LaTeX formatting with $$ for display equations
3. Added clear section headers
4. Preserved the Python-style pseudocode block
5. Included explanatory note about the * operator usage