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TR24-027
| 18th February 2024
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Dor Minzer, Kai Zhe Zheng#### Near Optimal Alphabet-Soundness Tradeoff PCPs

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TR24-026
| 15th February 2024
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Pavel Hrubes#### A subquadratic upper bound on sum-of-squares compostion formulas

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TR24-025
| 13th February 2024
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Mason DiCicco, Vladimir Podolskii, Daniel Reichman#### Nearest Neighbor Complexity and Boolean Circuits

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Dor Minzer, Kai Zhe Zheng

We show that for all $\varepsilon>0$, for sufficiently large prime power $q\in\mathbb{N}$, for all $\delta>0$, it is NP-hard to distinguish whether a $2$-Prover-$1$-Round projection game with alphabet size $q$ has value at least $1-\delta$, or value at most $1/q^{1-\varepsilon}$. This establishes a nearly optimal alphabet-to-soundness tradeoff for $2$-query PCPs ... more >>>

Pavel Hrubes

For every $n$, we construct a sum-of-squares identitity

\[ (\sum_{i=1}^n x_i^2) (\sum_{j=1}^n y_j^2)= \sum_{k=1}^s f_k^2\,,\]

where $f_k$ are bilinear forms with complex coefficients and $s= O(n^{1.62})$. Previously, such a construction was known with $s=O(n^2/\log n)$.

The same bound holds over any field of positive characteristic.

Mason DiCicco, Vladimir Podolskii, Daniel Reichman

A nearest neighbor representation of a Boolean function $f$ is a set of vectors (anchors) labeled by $0$ or $1$ such that $f(x) = 1$ if and only if the closest anchor to $x$ is labeled by $1$. This model was introduced by Hajnal, Liu, and TurĂ¡n (2022), who studied ... more >>>

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