Algorithm calculates Hopf invariant for simplicial mappings.
problem Computing Hopf invariant for simplicial mappings.
method Proposed an algorithm based on Whitehead's integral formula.
result Algorithm successfully computes Hopf invariant.
Two complete knot invariants from diagrams, finite or infinite.
problem Classifying knots completely.
method Constructed two invariants from knot diagrams, finite or infinite.
result Finite set reveals knotting number.
Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of category theory. The invariances that a supervised learning algorithm possesses are …
New algorithm calculates S-invariants for links efficiently.
problem Cannot be done by standard spectral sequence methods.
method Developed an algorithm for S-invariants of links. result Demonstrated efficiency and applicability to sl(3)-link homology.
Efficient algorithms for WRT invariants of torus bundles using algebraic structures.
problem Computing topological invariants of 3-manifolds is generally intractable.
method Embedding skein algebra into symmetric subalgebra at roots of unity for polynomial-time classical computation and using quantum algorithms for exponential space advantage.
result Polynomial-time classical computation and quantum algorithms for WRT invariants of torus bundles.
The paper matches features in images using centro-affine invariants and heat flow.
problem Feature matching in images with invariant algorithms.
method Developed an invariant algorithm using centro-affine invariants and heat flow.
result The algorithm compares favorably with existing feature matching methods.
By using grid homology theory, we give an explicit algorithm for computing Ozsváth-Stipsicz-Szabó's Υ-invariant and the d-invariant of Dehn surgeries along knots in S3. As its application, we compute the two invariants for all prime knots with up to 11 crossings.
New algorithm uses conditionally invariant components to improve domain adaptation performance.
problem Improving domain adaptation performance when source and target data distributions differ.
method Conditionally invariant components (CICs) and importance-weighted conditional invariant penalty (IW-CIP) algorithm.
result New algorithm provides target risk guarantees and addresses label-flipping features.
The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invariants are parameterised by an integer r≥3. We resolve the question …
Rotation invariant algorithms fail on sparse problems even with noise.
problem Rotation invariant algorithms' suboptimality in sparse linear problems with noise.
method Lower bounds and trajectory analysis of optimization algorithms.
result Rotation invariant algorithms are suboptimal even with noise and many examples.
We consider a variant of online convex optimization in which both the instances (input vectors) and the comparator (weight vector) are unconstrained. We exploit a natural scale invariance symmetry in our unconstrained setting: the predictions of the optimal comparator are invariant under any linear transformation of th…
Efficient algorithm computes knot invariants quickly.
problem Computing finite type invariants efficiently for knots.
method Create look-up tables for subdiagrams indexed by dyadic intervals, then compute invariants in ildeO(n⌈2kceil) time. result Finite type invariants can be computed on an n-crossing knot in ildeO(n⌈2kceil) time, significantly faster than previous methods. We make policy optimization algorithms batch size-invariant by decoupling proximal and behavior policies.
problem Some policy optimization algorithms do not have batch size-invariance, leading to inefficiencies.
method We decouple the proximal policy from the behavior policy to achieve batch size-invariance.
result Our approach makes policy optimization algorithms more efficient and allows them to use stale data more effectively.
Invariants for surfaces up to rigid transformations, with a comeagre subset retrieval algorithm.
problem Identifying compact surfaces up to rigid transformations.
method Degree four polynomials in moments of delta function, effective inversion algorithm.
result Invariants and retrieval algorithm work on a comeagre subset of surfaces.
New algorithm learns invariant representations for robust neural networks.
problem Learning robust neural network representations that are invariant to certain factors.
method Causal perspective and distribution matching approach.
result Empirically, the algorithm achieves state-of-the-art performance on domain generalization.
We discuss the invariant classification of vacuum Kundt waves using the Cartan-Karlhede algorithm, and the upper bound on the number of iterations of the Karlhede algorithm to classify the vacuum Kundt waves. By choosing a particular coordinate system we partially construct the canonical coframe used in the classificat…
Study s-invariants from sl(3)-link homology, extending to other characteristics.
problem Deepen understanding of s-invariants from Khovanov's homology. method Use Mackaay-Vaz universal sl(3)-link homology, extend to other characteristics, use scanning algorithm for calculations. result Extend s-invariants to other characteristics, particularly p=3. Algorithm calculates quantum invariants of 3-manifolds with polynomial time complexity.
problem Computing quantum invariants from Tambara-Yamagami categories is #P-hard.
method Fixed-parameter tractable algorithm with first Betti number as parameter.
result Existence of FPT algorithm for Tambara-Yamagami invariants.
Dihedral linking invariant uses knot colorings to distinguish knots.
problem Distinguishing knots using knot colorings and linking numbers.
method Algorithm for computing linking numbers in dihedral branched covers.
result The dihedral linking invariant distinguishes more than 98% of prime knot pairs.
New algorithms identify invariant features for domain generalization.
problem Achieving robust models across unseen environments.
method Invariant-Feature Subspace Recovery (ISR) algorithms.
result ISR algorithms achieve provable domain generalization with fewer training environments.
Rotation invariant algorithms fail with hard labels sampled from sparse targets.
problem Rotation invariant algorithms fail to learn from hard labels sampled from sparse targets.
method Proving the excess risk of rotation invariant algorithms and proposing a simple non-rotation invariant algorithm.
result Rotation invariant algorithms incur an excess risk of $Ω\left(\frac{d-1}{n}
ight)$, while non-rotation invariant algorithms have an excess risk of $O\left(\frac{s\log d}{n}
ight).
Implemented Habegger-Lin algorithm for 4- and 5-component links.
problem Determining link-homotopy of links.
method Explicit computation of group actions and implementation of algorithm.
result Found new pairs of links not distinguishable by Milnor's invariants.
New link invariants from diagram colorings match link widths.
problem Defining link widths via diagram colorings.
method Colorings of link diagrams to define invariants and prove their equivalence to link widths.
result Invariants of link widths calculated algorithmically.
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
SenSeI ensures fair models by enforcing invariance on sensitive groups.
problem Ensuring fair machine learning models that respect sensitive groups.
method Designing a transport-based regularizer to enforce invariance on sensitive sets.
result Certifiably fair ML models trained using SenSeI achieve improved fairness metrics.
We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional algorithm for computing these invariants.
New AMP algorithms for rotationally invariant models with reduced complexity.
problem Signal estimation in generalized linear models with arbitrary spectral design matrices.
method Rotationally invariant approximate message passing (AMP) algorithms.
result Performance close to Vector AMP with significantly lower complexity.
Researchers confirm a relation between knot invariants and provide formulas for torus knots.
problem Confirming a relation between knot invariants and providing formulas.
method Explicit formulas and algorithms for certain ADO-invariants of torus knots obtained from the series invariant of knot complements.
result Explicit formulas and algorithms for certain ADO-invariants of torus knots.
New algorithm reduces online learning regret by exploiting historical invariances.
problem Stochastic non-stationary linear bandits with changing reward models.
method ISD-linUCB algorithm that learns invariances in reward model.
result Significant regret improvements in fast-changing environments with historical data.
The paper calculates knot invariants using Blanchfield forms and obstructs sliceness.
problem Computing and obstructing the sliceness of knots.
method Algorithmic computation of twisted signature invariants using twisted Blanchfield forms and satellite formulas.
result Illustrated algorithm for (2,q)-torus knots and obstruction of sliceness for certain knots. New algorithm for signal estimation in noisy matrix models.
problem Signal estimation in rectangular spiked matrix models with rotationally invariant noise.
method Orthogonal Approximate Message Passing (OAMP) algorithm for signal estimation.
result Optimal OAMP algorithm minimizes mean-squared error and achieves Bayes-optimal performance.
This paper provides a geometrical derivation of the Hybrid Minimum Principle (HMP) for autonomous hybrid systems whose state manifolds constitute Lie groups (G,⋆) which are left invariant under the controlled dynamics of the system, and whose switching manifolds are defined as smooth embedded time invariant subma…
Two metrics on a manifold are geodesically equivalent if sets of their unparameterized geodesics coincide. In this paper we show that if two left G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connecti…
This paper investigates domain generalization: How to take knowledge acquired from an arbitrary number of related domains and apply it to previously unseen domains? We propose Domain-Invariant Component Analysis (DICA), a kernel-based optimization algorithm that learns an invariant transformation by minimizing the diss…
Introduces new Wasserstein distances for more intrinsic metrics.
problem Improve metric for comparing distributions.
method Introduces RWp distances, designs algorithms for computation. result New distances are more intrinsic and computable.
The paper analyzes the tradeoffs between accuracy and invariance in learning representations.
problem Achieving both accuracy and invariance in machine learning models.
method Information theoretic analysis of classification and regression settings.
result Characterization of the accuracy and invariance achievable by any representation of the data.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
Algorithms compute invariants of 4-manifolds as branched covers.
problem Computing invariants of 4-manifolds as branched covers.
method Diagrammatic algorithms using tri-plane diagrams and permutations.
result Automated algorithm for computing Kjuchukova's homotopy-ribbon obstruction.
Invariant Causal Set Covering Machines avoid spurious associations.
problem Learning algorithms for rule-based models are vulnerable to spurious associations.
method Building on invariant causal prediction, propose Invariant Causal Set Covering Machines for conjunctions/disjunctions of binary-valued rules.
result The method can identify causal parents of a variable of interest in polynomial time.
FAIRM learns fair and generalizable models by enforcing invariance across different data distributions.
problem Addressing fairness and domain generalization in machine learning models under heterogeneous data.
method FAIRM is a training environment-based oracle that enforces invariance across different data distributions, providing theoretical guarantees and efficient algorithms for linear models.
result FAIRM achieves minimax optimal performance and outperforms existing methods in synthetic and MNIST data evaluations.
We propose a new algorithm for Dehn surgery problem, finding exceptional Dehn filling slopes for a given hyperbolic 3-manifold with a torus boundary, using a quantum invariant called "3D index". The invariant is defined using an ideal triangulation of the cusped 3-manifold. We test the algorithm for many examples.
We propose an algorithm which allows to derive the generalized Alexander polynomial invariants of knots and links with the help of the q,p-numbers, appearing in bosonic two-parameter quantum algebra. These polynomials turn into HOMFLY ones by applying special parametrization. The Jones polynomials can be also obtained …
Pulling back the weight system associated with the exceptional Lie algebra G_2 by a modification of the universal Vassiliev-Kontsevich invariant yields a link invariant; extending it to 3-nets, we derive a recursive algorithm for its evaluation.
Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invariants can be computed purely combinatorially by enumerating colourings on the edges of a triangulation T. These edge colourings can be interpre…
In this short note, we observe that the Heegaard Floer contact invariant is combinatorial by applying the algorithm of Sarkar--Wang to the description of the contact invariant due to Honda--Kazez--Matic. We include an example of this combinatorial calculation.
Generalized algorithm for translation and scale-invariant prediction.
problem Sequential prediction with expert advice, focusing on translation and scale invariance.
method Designing a generalized online algorithm using the universal prediction perspective to compete against a generic class of expert selection strategies.
result No preliminary knowledge of loss sequences is required; performance bounds are stable under arbitrary scalings and translations.
We use the divide-and-conquer and scanning algorithms for calculating Khovanov cohomology directly on the Lee- or Bar-Natan deformations of the Khovanov complex to give an alternative way to compute Rasmussen s-invariants of knots. By disregarding generators away from homological degree 0 we can considerably improve …
Pulling back the weight system associated with the spinor representation of the Lie algebra so(7) by the universal Vassiliev-Kontsevich invariant yields a numerical link invariant with values in formal power series. Computing some skein relations satisfied by this invariant, I derive a recursive algorithm for its evalu…