Core Concepts
We prove a new lifting theorem that works for every two functions f and g such that the discrepancy of g is at most inverse polynomial in the input length of f. This significantly generalizes the known direct-sum theorem for discrepancy and extends the range of inner functions g for which lifting theorems hold.
Abstract
The content discusses lifting theorems, which are theorems that bound the communication complexity of a composed function f ◦ g^n in terms of the query complexity of f and the communication complexity of g. Such theorems constitute a powerful generalization of direct-sum theorems for g, and have seen numerous applications in recent years.
The main contribution of the content is a new lifting theorem that works for every two functions f and g such that the discrepancy of g is at most inverse polynomial in the input length of f. This significantly generalizes the known direct-sum theorem for discrepancy, and extends the range of inner functions g for which lifting theorems hold.
The authors first provide background on lifting theorems and their relation to direct-sum theorems. They then state their main theorem, which shows that if the discrepancy of g is at least logarithmic in the input length of f, then the communication complexity of f ◦ g^n is lower bounded by the product of the query complexity of f and the discrepancy of g.
The authors then discuss the techniques used to prove this result, focusing on the main technical lemma. This lemma extends the main lemma of a previous work by [CFK+19], which was limited to the case where the discrepancy of g is at least linear in the input length of g. The authors overcome this limitation by introducing the notion of "recoverable" values, which allows them to handle the case where the discrepancy of g is smaller.
Finally, the authors discuss some related work and open questions, including the conjecture that lifting theorems should hold for every inner function g that has a sufficiently large information cost.
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