Option prices encode the market's collective outlook through implied density and implied volatility. An explicit link between implied density and implied volatility translates the risk-neutrality of the former into conditions on the latter to rule out static arbitrage. Despite earlier recognition of their parity, the two had been studied in isolation for decades until the recent demand in implied volatility modeling rejuvenated such parity. This paper provides a systematic approach to build neural representations of option implied information. As a preliminary, we first revisit the explicit link between implied density and implied volatility through an alternative and minimalist lens, where implied volatility is viewed not as volatility but as a pointwise corrector mapping the Black-Scholes quasi-density into the implied risk-neutral density. Building on this perspective, we propose the neural representation that incorporates arbitrage constraints through the differentiable corrector. With an additive logistic model as the synthetic benchmark, extensive experiments reveal that deeper or wider network structures do not necessarily improve the model performance due to the nonlinearity of both arbitrage constraints and neural derivatives. By contrast, a shallow feedforward network with a single hidden layer and a specific activation effectively approximates implied density and implied volatility.

We study fairness in collaborative-filtering recommender systems, which are sensitive to discrimination that exists in historical data. Biased data can lead collaborative-filtering methods to make unfair predictions for users from minority groups. We identify the insufficiency of existing fairness metrics and propose four new metrics that address different forms of unfairness. These fairness metrics can be optimized by adding fairness terms to the learning objective. Experiments on synthetic and real data show that our new metrics can better measure fairness than the baseline, and that the fairness objectives effectively help reduce unfairness.

Cognitive processes usually take place at a macroscopic scale in systems charac-terised by emergent properties that make the whole'more than the sum of its parts.'

This work investigates how these complexities necessarily arise for feature learning in the presence of computational-statistical gaps.