Topic
Bilinear form
About: Bilinear form is a research topic. Over the lifetime, 3538 publications have been published within this topic receiving 68271 citations.
Papers published on a yearly basis
1 / 2
Papers
7 Dec 2015
TL;DR: Blinear models, a recognition architecture that consists of two feature extractors whose outputs are multiplied using outer product at each location of the image and pooled to obtain an image descriptor, are proposed.
Abstract: We propose bilinear models, a recognition architecture that consists of two feature extractors whose outputs are multiplied using outer product at each location of the image and pooled to obtain an image descriptor. This architecture can model local pairwise feature interactions in a translationally invariant manner which is particularly useful for fine-grained categorization. It also generalizes various orderless texture descriptors such as the Fisher vector, VLAD and O2P. We present experiments with bilinear models where the feature extractors are based on convolutional neural networks. The bilinear form simplifies gradient computation and allows end-to-end training of both networks using image labels only. Using networks initialized from the ImageNet dataset followed by domain specific fine-tuning we obtain 84.1% accuracy of the CUB-200-2011 dataset requiring only category labels at training time. We present experiments and visualizations that analyze the effects of fine-tuning and the choice two networks on the speed and accuracy of the models. Results show that the architecture compares favorably to the existing state of the art on a number of fine-grained datasets while being substantially simpler and easier to train. Moreover, our most accurate model is fairly efficient running at 8 frames/sec on a NVIDIA Tesla K40 GPU. The source code for the complete system will be made available at http://vis-www.cs.umass.edu/bcnn.
2,435 citations
TL;DR: The main theorem gives an error estimate in terms of localized quantities which can be computed approximately, and the estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same.
Abstract: A mathematical theory is developed for a class of a-posteriors error estimates of finite element solutions. It is based on a general formulation of the finite element method in terms of certain bilinear forms on suitable Hilbert spaces. The main theorem gives an error estimate in terms of localized quantities which can be computed approximately. The estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same. The theoretical results also lead to a heuristic characterization of optimal meshes, which in turn suggests a strategy for adaptive mesh refinement. Some numerical examples show the approach to be very effective.
1,560 citations
Book•
1 Dec 1984
TL;DR: In this article, the authors present a classification of Quadratic Forms over Fields and their application in Fields of Prescribed Level, including the following: 1. Symmetric Bilinear Forms over Dedekind rings and Global Fields.
Abstract: 1. Basic Concepts.- 1. Bilinear Forms and Quadratic Forms.- 2. Matrix Notation.- 3. Regular Spaces and Orthogonal Decomposition.- 4. Isotropy and Hyperbolic Spaces.- 5. Witt's Theorem.- 6. Appendix: Symmetric Bilinear Forms and Quadratic Forms over Rings.- 2. Quadratic Forms over Fields.- 1. Grothendieck and Witt Rings.- 2. Invariants.- 3. Examples I (Finite Fields).- 4. Examples II (Ordered Fields).- 5. Ground Field Extension and Transfer.- 6. The Torsion of the Witt Group.- 7. Orderings, Pfister's Local Global Principle, and Prime Ideals of the Witt Ring.- 8. Applications of the Method of Transfer.- 9. Description of the Witt Ring by Generators and Relations.- 10. Multiplicative Forms.- 11. Quaternion Algebras.- 12. The Hasse Invariant and the Witt Invariant.- 13. The Hasse Algebra.- 14. Classification Theorems.- 15. Examples III. Ci-fields.- 16. The u-invariant.- 3. Quadratic Forms over Formally Real Fields.- 1. Formally Real and Ordered Fields.- 2. Real Closed Fields.- 3. Hilbert's 17th Problem and the Real Nullstellensatz.- 4. Extension of Signatures.- 5. The Space of Orderings of a Field.- 6. The Total Signature.- 7. A Local Global Principle for Weak Isotropy.- Appendix: Places, Valuations, and Valuation Rings.- 4. Generic Methods and Pfister Forms.- 1. Chain-p-equivalence of Pfister Forms.- 2. Pfister's Theorem on the Representation of Positive Functions as Sums of Squares.- 3. Casseis' and Pfister's Representation Theorems.- 4. Applications: Fields of Prescribed Level. Characterization of Pfister Forms.- 5. The Function Field of a Quadratic Form and the Main Theorem of Arason and Pfister.- 6. Generic Zeros and Generic Splitting.- 7. Knebusch's Filtration of the Witt Ring.- 5. Rational Quadratic Forms.- 1. Symmetric Bilinear Forms and Quadratic Forms on Finite Abelian Groups.- 2. Gaussian Sums for Quadratic Forms on Finite Abelian Groups.- 3. The Witt Group of1.- 4. The Witt Group of 2.- 5. Gauss' First Proof of the Quadratic Reciprocity Law.- 6. Quadratic Forms over the p-adic Numbers.- 7. Hilbert's Reciprocity Law and the Hasse-Minkowski Theorem.- 8. Calculation of Gaussian Sums.- 6. Symmetric Bilinear Forms over Dedekind Rings and Global Fields.- 1. Symmetric Bilinear Forms over Dedekind Rings.- 2. Symmetric Bilinear Forms over Discrete Valuation Rings.- 3. Symmetric Bilinear Forms over Polynomial Rings and Rational Function Fields.- 4. Symmetric Bilinear Forms over p-adic Fields.- 5. The Hilbert Reciprocity Theorem.- 6. The Hasse-Minkowski Theorem.- 7. Hecke's Theorem on the Different.- 8. The Residue Theorem.- 7. Foundations of the Theory of Hermitian Forms.- 1. Basic Definitions.- 2. Hermitian Categories.- 3. Quadratic Forms.- 4. Transfer and Reduction.- 5. Hermitian Abelian Categories.- 6. Hermitian Forms over Skew Fields.- 7. Hyperbolic Forms and the Unitary Group.- 8. Alternating Forms and the Symplectic Group.- 9. Witt's Theorem.- 10. The Krull-Schmidt Theorem.- 11. Examples and Applications.- 8. Simple Algebras and Involutions.- 1. Simple Rings and Modules.- 2. Tensor Products.- 3. Central Simple Algebras. The Brauer Group.- 4. Simple Algebras.- 5. Central Simple Algebras under Field Extensions. Reduced Norms and Traces.- 6. Examples.- 7. Involutions on Simple Algebras. The Classification Problem.- 8. Existence of Involutions.- 9. The Corestriction. Existence of Involutions of the Second Kind.- 10. An Extension Theorem for Involutions.- 11. Quaternion Algebras.- 12. Cyclic Algebras.- 13. The Canonical Involution on the Group Algebra.- 9. Clifford Algebras.- 1. Graded Algebras.- 2. Clifford Algebras.- 3. The Spinor Norm.- 4. Quadratic Forms over Fields in Characteristic 2.- 10. Hermitian Forms over Global Fields.- 1. Hermitian Forms over Commutative Fields and Quaternion Algebras.- 2. Simple Algebras and Involutions over Local and Global Fields.- 3. Skew Hermitian Forms over Quaternion Fields.- 4. Skew Hermitian Forms over Global Quaternion Fields..- 5. The Strong Approximation Theorem.- 6. Hermitian Forms for Unitary Involutions. Statement of Results.- 7. Proof of the Weak Local Global Principle.- 8. Conclusion of the Proof.
1,302 citations
1 Jan 1992
TL;DR: In this paper, a functional analytic background on Markov processes and Dirichlet forms is provided, together with an analysis of the potential theory of Dirichlets and its application to Markov Processes.
Abstract: 0 Introduction.- I Functional Analytic Background.- 1 Resolvents, semigroups, generators.- 2 Coercive bilinear forms.- 3 Closability.- 4 Contraction properties.- 5 Notes/References.- II Examples.- 1 Starting point: operator.- 2 Starting point: bilinear form - finite dimensional case.- 3 Starting point: bilinear form - infinite dimensional case.- 4 Starting point: semigroup of kernels.- 5 Starting point: resolvent of kernels.- 6 Notes/References.- III Analytic Potential Theory of Dirichlet Forms.- 1 Excessive functions and balayage.- 2 ?-exceptional sets and capacities.- 3 Quasi-continuity.- 4 Notes/References.- IV Markov Processes and Dirichlet Forms.- 1 Basics on Markov processes.- 2 Association of right processes and Dirichlet forms.- 3 Quasi-regularity and the construction of the process.- 4 Examples of quasi-regular Dirichlet forms.- 5 Necessity of quasi-regularity and some probabilistic potential theory.- 6 One-to-one correspondences.- 7 Notes/References.- V Characterization of Particular Processes.- 1 Local property and diffusions.- 2 A new capacity and Hunt processes.- 3 Notes/References.- VI Regularization.- 1 Local compactification.- 2 Consequences - the transfer method.- 3 Notes/References.- A Some Complements.- 1 Adjoint operators.- 2 The Banach/Alaoglu and Banach/Saks theorems.- 3 Supplement on Ray resolvents and right processes.
1,247 citations
Book•
1 Jan 1973
TL;DR: In this article, the Hasse-Minkowski Theorem and the Signature mod 8 of the Quadratic Reciprocity Theorem are used to describe the inner product spaces over a field.
Abstract: I. Basic Concepts.- II. Symmetric Inner Product Spaces over Z.- III. Inner Product Spaces over a Field.- IV. Discrete Valuations and Dedekind Domains.- V. Some Examples.- Appendix 1. Quadratic Forms.- Appendix 2. Hermitian Forms.- Appendix 3. The Hasse-Minkowski Theorem.- Appendix 4. Gauss Sums, the Signature mod 8, and Quadratic Reciprocity.- Appendix 5. The Leech Lattice, and Other Lattices in Dimension 24.- Chronological Table.- References.- Special Notations.
1,090 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 8 |
| 2024 | 29 |
| 2023 | 104 |
| 2022 | 212 |
| 2021 | 171 |
| 2020 | 180 |