Min-Max Normalization

The core operation maps every score x to a new value x' using the formula: x' = (x - min) / (max - min). This preserves the original distribution's shape and relative distances between scores. The minimum observed score maps to 0.0, and the maximum maps to 1.0. This is a crucial step before combining BM25 and vector search results, as raw cosine similarity scores and term-frequency scores exist on entirely different numerical scales.

A critical vulnerability exists in the algorithm's reliance on the observed min and max values. A single extreme outlier in the result set can compress the rest of the normalized scores into a tiny sub-range near 0.0 or 1.0, destroying score differentiation. To mitigate this, robust implementations often pre-compute theoretical min/max bounds or use winsorization to cap extreme values at a specific percentile (e.g., the 99th) before applying the normalization formula.

Unlike non-linear methods such as rank-based normalization, Min-Max preserves the exact distribution shape of the original scores. If the raw dense retrieval scores are tightly clustered around 0.85, the normalized scores will remain tightly clustered. This property is vital when the magnitude of the score difference carries semantic meaning, such as indicating a high-confidence match versus a marginal one, rather than just a relative ordering.

This technique is a fundamental preprocessing step for Weighted Sum Fusion and CombSUM. Without rescaling, a BM25 score of 25.0 would mathematically dominate a cosine similarity score of 0.92 in a linear combination, effectively silencing the semantic signal. By projecting both scores to [0, 1], a weighted sum like 0.4 * BM25_norm + 0.6 * Dense_norm becomes a meaningful, tunable balance between lexical precision and semantic understanding.

A mathematical failure mode occurs when all retrieved documents have the exact same score, causing max - min to equal zero. A direct division by zero is undefined. Production systems must include a guard clause: if the range is zero, the function should return a constant value (e.g., 0.5 or 1.0) for all items, effectively treating the retrieval signal as non-discriminative and preventing a runtime exception in the search pipeline.

The definition of the min and max depends entirely on the context. In a batch re-ranking scenario, min and max are calculated from the current candidate pool. In a streaming or production indexing context, min and max must be derived from a statistically significant historical sample or a known theoretical range of the scoring function (e.g., cosine similarity is naturally bounded between -1 and 1). Using the wrong context leads to unstable normalization.