咨询：选择5个列向量以最大化点积的启发式或线性代数方法

Hey there! Let's break down your problem and walk through some practical, fast approaches that don't require fancy optimization tool expertise. First, let's rephrase the problem in simpler terms to make it easier to tackle:

When you take the Hadamard product of 5 columns, the resulting strategy vector has a 1 in row i only if all 5 columns have a 1 in row i. The dot product with your ±1 target vector is just:

(number of rows where all 5 columns are 1 AND target value is 1) - (number of rows where all 5 columns are 1 AND target value is -1)

With that in mind, here are your best bets for fast, effective solutions:

Heuristic Methods (Fast & Easy to Implement) #

These methods prioritize speed over perfect optimality, which aligns with your goal of avoiding multi-month/year runtimes.

1. Greedy Iterative Selection #

This is probably the most straightforward and fast approach:

• Step 1: Precompute a "base score" for every column: calculate the dot product of the column with your target vector (this is just the count of 1s in the column where target is 1 minus the count where target is -1). Pick the column with the highest base score as your first selection.
• Step 2: For each remaining column, calculate the score you'd get by adding it to your current set (i.e., the dot product of the Hadamard product of all selected columns so far plus this new column, with the target vector). Pick the column that gives the biggest score boost.
• Step 3: Repeat step 2 until you have 5 columns.
• Pro tip for sparse matrices: Store each column's 1-row indices as a set (since sparse matrices have far fewer 1s than 0s). To compute the score for adding a new column, just find the intersection of the current selected columns' 1-row sets with the new column's set, then sum the target values for those rows. This is way faster than iterating through all 60,000 rows.

2. Positive-Focused Random Sampling with Filtering #

If you want to balance speed with exploring more combinations:

• Step 1: Filter down your columns to a smaller candidate pool first. Focus on columns that have lots of 1s in rows where the target vector is 1 (since these contribute positively to your score). For example, keep the top 200 columns with the highest count of 1s in positive target rows.
• Step 2: Randomly sample hundreds of 5-column combinations from this pool (1000-5000 samples is enough). Calculate the score for each combination and keep the highest-scoring one.
• Why this works: By narrowing to a high-quality pool, you avoid wasting time on columns that can't contribute to a good score. Random sampling lets you explore more combinations than greedy alone, without the computational cost of checking every possible combination.

3. Penalty-Avoidance Heuristic #

If your target vector has a lot of -1 values (penalties), add this step before using the above methods:

• Filter out columns that have a high number of 1s in rows where the target is -1. For example, discard any column where more than 30% of its 1s are in penalty rows. This reduces the chance of your 5-column combination having overlapping 1s in penalty rows, which would drag down your score.

Linear Algebra-Based Fast Tricks #

Leverage your sparse matrix structure to speed up calculations:

• Precompute a modified target vector w where each element is t_i (your original target value). For any column c, create a vector c_w where each element is c_i * w_i (so 0 stays 0, 1 becomes t_i).
• The score for a set of columns is the sum of c_w values where all columns have a 1. For sparse matrices, this is equivalent to finding the intersection of the columns' 1-row indices and summing w at those indices—using numpy or set operations makes this lightning fast.
• If you're using scipy.sparse (even if you don't want to use its optimization tools), storing your matrix in CSC format lets you quickly extract the 1-row indices for any column using M.indptr and M.indices.

Tool Recommendations (Google OR-Tools & Scipy) #

You mentioned you don't have experience with these tools—here's the straight scoop:

• Scipy Optimization: Most of Scipy's tools are for continuous optimization, which doesn't fit your discrete column-selection problem. Its discrete optimization support is limited, and setting it up would take more time than just implementing the heuristics above.
• Google OR-Tools: It has combinatorial optimization solvers that could model this problem as an integer program, but with 10,000 columns to choose from, the solver would struggle to find a solution quickly (if at all). You'd also need to learn how to model the problem, which is more work than the heuristic approaches.
• Bottom line: Stick with the heuristic methods—they're faster, easier to implement, and give you good enough results without the learning curve.

Quick Implementation Tips #

• Use sparse matrix formats (like CSC from scipy.sparse) to avoid wasting memory and speed up column operations.
• Store each column's 1-row indices as a set or sorted numpy.ndarray—set intersections are optimized in Python and will save you tons of time.
• When calculating scores for combinations, use vectorized operations with numpy (e.g., t[intersection_rows].sum()) instead of looping through rows—this is orders of magnitude faster.

内容的提问来源于stack exchange，提问作者Alex Pharaon