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):
```python
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
