arXiv research

Paper solves k-sparse parity problem with sign SGD, matching SQ lower bound.

SGD learns sparse parities near computational limits with discontinuous phase transitions.

This paper shows neural networks can learn non-linear sparse parities.

Curriculum learning helps neural networks learn parities more efficiently.

This work explores feature learning tradeoffs in neural networks.

Solves a model for sudden problem-solving ability in deep learning.

Polynomial-time algorithm solves random parity games with high probability.

Extracting a curriculum from a teacher network improves distillation efficiency.

Parity functors assign labels to knot diagrams based on crossing parity.

Parity defined for based matrices, a new example of virtual knot parity.

We introduce the 2-colour parity. It is a theory of parity for a large class of virtual links, defined using the interaction between orientations of the link components and a certain type of colouring. The 2-colour parity is an extension of the Gaussian parity, to which it reduces on virtual knots. We show that the 2-c…

In the present paper, we develop the parity theory invented in \cite{ManSb}; we construct new parities for two-component (virtual and free) links. New parities significantly depend on geometrical properties of diagrams; in particular, they are mutation-sensitive. New parities can be used practically in all problems, wh…

In \cite {FrKn,Sbornik} it was shown that in some knot theories the crucial role is played by {\em parity}, i.e.\ a function on crossings valued in $\{0,1\}$ and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and …

We identify conditional parity as a general notion of non-discrimination in machine learning. In fact, several recently proposed notions of non-discrimination, including a few counterfactual notions, are instances of conditional parity. We show that conditional parity is amenable to statistical analysis by studying ran…

Universal Gaussian parity proven for 2D knots.

New parities defined on virtual knots linked to crossing indices.

Derives FACT, an alternative to NFA for neural networks, explaining feature learning.

This paper tackles fair Bayes-optimal classifiers under predictive parity, proving their limitations and proposing a new algorithm.

Parity calibration aims to predict increase-decrease events, not values.

We use crossing parity to construct a generalization of biquandles for virtual knots which we call Parity Biquandles. These structures include all biquandles as a standard example referred to as the even parity biquandle. Additionally, we find all Parity Biquandles arising from the Alexander Biquandle and Quaternionic …

Diversified risk parity strategies outperform equally-weighted portfolios in various asset universes.

We learn higher-order Markov random fields from evolving data, bypassing computational barriers.

CoT improves transformer sample efficiency by reducing input token dependencies and attention sparsity.

Functorial maps and weak parities are equivalent descriptions of rules of substitution virtual crossings for classical in diagrams of a knot in a way compatible with Reidemeister moves. We introduce the notion of maximal weak parity and describe it for knots in a given closed oriented surface. This weak parity defines …

Introduce a two-variable parity polynomial for virtual knotoids

Neural networks struggle with learning fixed parities.

In [3] we constructed the parity-biquandle bracket valued in {\em pictures} (linear combinations of $4$-valent graphs). We gave no example of classical links such that the parity-biquandle bracket of which is not trivial. In the present paper we slightly change the notation of the parity-biquandle bracket and give exam…

Study shows physical drift affects put-call parity enforcement, not just option payoffs.

Parity mappings from the chords of a Gauss diagram to the integers is defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and virtual knots. One family consists of degree $n$ Vassiliev invariants.

We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. These invariants are determined by pairs consisting of a biquandle 2-cocycle φ^0 and a map φ^1 with certain compatibility conditions leading to one-variable or two-variable polynomial invariants of virtual knots. We provide …

Transformers can generalize to a large task family with only a few demonstrations.

Proposes a method to achieve quantile fairness in predictions.

Counterfactual fairness not equivalent to demographic parity, finds study.

Transformers solve parity problems efficiently with step-by-step reasoning.

2-dimensional knots and links are studied in the article. The notion of parity is introduced via techniques similar to the ones used by the second named author in 1-dimensional case. By using parity new invariants are constructed and known invariants are refined.

Classical clients can verify quantum learning tasks efficiently.

TRP uses tree-based approach for market-neutral portfolios.

Transformers learn sparse Boolean functions through RL and SFT, revealing distinct learning behaviors.

New causal analysis reconciles predictive and statistical fairness.

Proposes a framework to create fair IDRs by enforcing demographic parity constraints.

New method controls bias in training data for fair outcomes.

The article develops a model for skewness risk in risk parity portfolios.

A new iterative low complexity algorithm has been presented for computing the Walsh-Hadamard transform (WHT) of an $N$ dimensional signal with a $K$-sparse WHT, where $N$ is a power of two and $K = O(N^α)$, scales sub-linearly in $N$ for some $0 < α< 1$. Assuming a random support model for the non-zero transform domain…

We mathematically compare four competing definitions of group-level nondiscrimination: demographic parity, equalized odds, predictive parity, and calibration. Using the theoretical framework of Friedler et al., we study the properties of each definition under various worldviews, which are assumptions about how, if at a…

We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads to a possibility of constructing new invariants and proving minimality and non-t…

We investigate an application of crossing parity for the bracket expansion of the Jones polynomial for virtual knots. In addition we consider an application of parity for the arrow polynomial as well as for the categorifications of both polynomials. We present a number of examples found through our calculations. We pro…

Neural networks outperform NTK on compositional tasks, revealing a complexity gap.

This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial. The paper is relatively self-contained and it describes virtual knot theory bot…