Learned Functional Projection: Learning for Recovery in Implied Volatility Smoothing

Implied volatility smoothing requires recovering continuous surfaces from sparse, irregular market quotes, yet approximation quality alone does not determine recovery accuracy. We prove that representations with zero approximation error can have arbitrarily large recovery-risk ratios under the same observations and coefficient-fitting rule. This motivates selecting representations by their recovery risk under the available observations. We introduce Learned Functional Projection (LFP), which learns a low-dimensional continuous function space by optimizing recovery from partial quotes. Training differentiates through a weighted ridge solve and evaluates predictions on held-out quotes; each new surface requires only one explicit coefficient solve. Financial regularization penalizes shape violations. The explicit recovery rule admits exact error identities that separate approximation, regularization and input error, together with a deterministic pointwise upper bound and a representation-error lower bound on recovery risk. Experiments use 2012–2021 data from four equity-index option markets. On SPX under standard inputs, LFP reduces held-out vega-weighted relative RMSE by 61.8% against the graph neural operator (GNO). With the same nine reference quotes, LFP-9 reduces this error by 58.4% against HyperIV and achieves 3.9 times its bid–ask coverage. The SPX-trained representation also leads all evaluated baselines in quote accuracy on NDX and RUT without retraining. Surface construction takes milliseconds, with approximately $42\times$ CPU and $113\times$ GPU speedups over the public GNO implementation. Controlled ablations show that learning for recovery outperforms learning for reconstruction, supporting the method's design and the practical relevance of the theoretical distinction.