Related papers: Dandelion: Certified Approximations of Elementary …
Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were…
The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The…
We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function $f$ is computable by a 1-query quantum algorithm with error bounded by $\epsilon<1/2$ iff $f$ can be…
Homomorphic encryption is one of the representative solutions to privacy-preserving machine learning (PPML) classification enabling the server to classify private data of clients while guaranteeing privacy. This work focuses on PPML using…
Electronic exams (e-exams) have the potential to substantially reduce the effort required for conducting an exam through automation. Yet, care must be taken to sacrifice neither task complexity nor constructive alignment nor grading…
Achieving faster execution with shorter compilation time can foster further diversity and innovation in neural networks. However, the current paradigm of executing neural networks either relies on hand-optimized libraries, traditional…
Mathematics formalisation is the task of writing mathematics (i.e., definitions, theorem statements, proofs) in natural language, as found in books and papers, into a formal language that can then be checked for correctness by a program. It…
We study the problem of {\sl certification}: given queries to a function $f : \{0,1\}^n \to \{0,1\}$ with certificate complexity $\le k$ and an input $x^\star$, output a size-$k$ certificate for $f$'s value on $x^\star$. This abstractly…
The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. We introduce a generic method for increasing the…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest…
Epistemic protocol specifications allow programs, for settings in which multiple agents act with incomplete information, to be described in terms of how actions are related to what the agents know. They are a variant of the knowledge-based…
The floating-point implementation of a function on an interval often reduces to polynomial approximation, the polynomial being typically provided by Remez algorithm. However, the floating-point evaluation of a Remez polynomial sometimes…
On the one hand, ordered completion is a fundamental technique in equational theorem proving that is employed by automated tools. On the other hand, their complexity makes such tools inherently error prone. As a remedy to this situation we…
We present FormalProofBench, a private benchmark designed to evaluate whether AI models can produce formally verified mathematical proofs at the graduate level. Each task pairs a natural-language problem with a Lean~4 formal statement, and…
Model counting, or counting the satisfying assignments of a Boolean formula, is a fundamental problem with diverse applications. Given #P-hardness of the problem, developing algorithms for approximate counting is an important research area.…
We present an algorithm for efficient evaluation of Boys functions $F_0,\dots,F_{k_\mathrm{max}}$ tailored to modern computing architectures, in particular graphical processing units (GPUs), where maximum throughput is high and data…
We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a dierence between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at…
A statistical emulator can be used as a surrogate of complex physics-based calculations to drastically reduce the computational cost. Its successful implementation hinges on an accurate representation of the nonlinear response surface with…
This paper presents an algorithm, Voted Kernel Regularization , that provides the flexibility of using potentially very complex kernel functions such as predictors based on much higher-degree polynomial kernels, while benefitting from…