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Matrix property in linear algebra
In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors
Restricted_isometry_property
Distance-preserving mathematical transformation
element of the domain. Note that ε-isometries are not assumed to be continuous. The restricted isometry property characterizes nearly isometric matrices
Isometry
Australian and American mathematician (born 1975)
the Gaussian ensemble, a large number of matrices satisfy the restricted isometry property.[CT06] In 2007, Candes and Tao introduced a novel statistical
Terence_Tao
"nullspace property" originates from Cohen, Dahmen, and DeVore. The nullspace property is often difficult to check in practice, and the restricted isometry property
Nullspace_property
Algorithm that estimates unknowns from a series of measurements over time
from the theory of compressed sensing/sampling, such as the restricted isometry property and related probabilistic recovery arguments, for sequentially
Kalman_filter
Mathematical result
constructions are suggested to circumvent this. Random projection Restricted isometry property Word embeddings For instance, writing about nearest neighbor
Johnson–Lindenstrauss_lemma
Means to measure signal processing ability
satisfy certain specific conditions such as RIP (Restricted Isometry Property) or Null-Space property in order to achieve robust sparse recovery. In the
Detection_theory
Method for solving certain optimization problems
convergence for ℓ1 norm and superlinear for ℓt with t < 1, under the restricted isometry property, which is generally a sufficient condition for sparse solutions
Iteratively reweighted least squares
Iteratively_reweighted_least_squares
Topics referred to by the same term
Regulation of Investigatory Powers Act 2000, a UK act of Parliament Restricted isometry property, in mathematics R.I.P. cartridge (Round, Irritant, Personnel)
RIP_(disambiguation)
Concept in mathematics
(using the spark (mathematics), the mutual coherence or the restricted isometry property) and the level of sparsity in the solution, k {\displaystyle
Sparse_approximation
Similar to the basis of a vector space, but not necessarily linearly independent
f_{y}\rangle d\mu (x).} k-frame Biorthogonal wavelet Orthogonal wavelet Restricted isometry property Schauder basis Harmonic analysis Fourier analysis Functional
Frame_(linear_algebra)
invariant subspaces of V. So V(K2) = K2. In other words, V restricted to K2 is a surjective isometry, i.e., a unitary operator U. Furthermore, each Mi is isomorphic
Wold's_decomposition
Lie group of Lorentz transformations
form a composition algebra. The isometry property of Lorentz transformations holds according to the composition property | p q | = | p | × | q | {\displaystyle
Lorentz_group
Fewest dependent columns in a matrix
(November 8, 2013). "The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing". IEEE
Spark_(mathematics)
Statistical analysis technique
Pfetsch (2013). "The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing". IEEE
Sparse_PCA
Filling in missing entries of a matrix
small. Here the matrix completion problem does not obey the restricted isometry property (RIP). For matrices, the RIP would assume that the sampling operator
Matrix_completion
Neural network coding model
interest. Also included are the concepts of mutual coherence and restricted isometry property to establish uniqueness stability guarantees. Allow signal x
Convolutional_sparse_coding
Groups of point isometries in 3 dimensions
in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a
Point groups in three dimensions
Point_groups_in_three_dimensions
Theorem about admissible crystal symmetries
isometries of finite order are of order 1, 2, 3, 4 or 6. Isometries of order n include, but are not restricted to, n-fold rotations. The theorem also excludes
Crystallographic restriction theorem
Crystallographic_restriction_theorem
Polynomial with all terms of degree two
T : V → V′ (isometry) such that Q ( v ) = Q ′ ( T v ) for all v ∈ V . {\displaystyle Q(v)=Q'(Tv){\text{ for all }}v\in V.} The isometry classes of n-dimensional
Quadratic_form
Value in matrix theory
bound can be constructed by Weil's theorem. Compressed sensing Restricted isometry property Babel function Tropp, J.A. (March 2006). "Just relax: Convex
Mutual coherence (linear algebra)
Mutual_coherence_(linear_algebra)
Signal-processing paradigm that trades precision for volume of measurements
for Dithering: Fast and Quantized Random Embeddings via the Restricted Isometry Property". Information and Inference. 6 (4): 441–476. arXiv:1607.00816
Sample_abundance
Feature of a system that is preserved under some transformation
spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed
Symmetry_(physics)
Vector space with generalized dot product
for all x ∈ V . {\displaystyle x\in V.} A linear isometry (resp. an antilinear isometry) is an isometry that is also a linear map (resp. an antilinear map)
Inner_product_space
Strong form of uniform continuity
distance between all points Dini continuity Modulus of continuity Quasi-isometry Johnson–Lindenstrauss lemma – For any integer n ≥ 0 {\displaystyle n\geq
Lipschitz_continuity
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
preserves the metric. Flows generated by Killing vector fields are continuous isometries of the manifold. This means that the flow generates a symmetry, in the
Killing_vector_field
Type of matrix representation
an isometry when its action is restricted onto the support of A {\displaystyle A} , that is, it means that U {\displaystyle U} is a partial isometry. As
Polar_decomposition
Group of symmetries of a regular polygon
multiples of 36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors
Dihedral_group
Formula relating the norm and the inner product in an inner product space
other point by a linear isometry, then the norm is induced by an inner product. The polarization identities are not restricted to inner products. If B
Polarization_identity
Mathematical space with a notion of distance
bijective distance-preserving function is called an isometry. One perhaps non-obvious example of an isometry between spaces described in this article is the
Metric_space
Blue tectosilicate mineral
this structure the two cavities are still chiral, because no indirect isometry centred on the cavity (i.e. a reflexion, inversion, or improper rotation)
Sodalite
Theorem in harmonic analysis
belongs to L 2 {\displaystyle L^{2}} , and the Fourier transform is an isometry with respect to the L2 norm, which is to say that ∫ − ∞ ∞ | f ( x ) | 2
Plancherel_theorem
Mathematical invariance under transformations
Automorphism Burnside's lemma Chirality Even and odd functions Fixed points of isometry groups in Euclidean space – center of symmetry Isotropy Palindrome Spacetime
Symmetry
Geometric transformation that preserves lines but not angles nor the origin
group. A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance. Each of these groups has a
Affine_transformation
Multidimensional data algorithm
shown to have stability and performance guarantees under certain restricted isometry conditions. The incremental multi-parameter algorithm (IMP), published
Matching_pursuit
Manifold with inversion symmetry
manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces
Hermitian_symmetric_space
Matrix group
( d , 1 ) , d ⩾ 2 {\displaystyle \mathrm {Sp} (d,1),d\geqslant 2} (the isometry groups of a sesquilinear form over the Hamilton quaternions), plus the
Congruence_subgroup
Product of the principal curvatures of a surface
surface S in R3. A local isometry is a diffeomorphism f : U → V between open regions of R3 whose restriction to S ∩ U is an isometry onto its image. Theorema
Gaussian_curvature
Rational function of the form (az + b)/(cz + d)
The Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying
Möbius_transformation
Azimuthal equal-area map projection
preserve both angles and areas. (If one did, then it would be a local isometry and would preserve Gaussian curvature; but the sphere and disk have different
Lambert azimuthal equal-area projection
Lambert_azimuthal_equal-area_projection
Non-orientable surface with one edge
MR 3370020. S2CID 119640200. Stillwell, John (1992). "4.6 Classification of isometries". Geometry of Surfaces. Universitext. Cham: Springer. pp. 96–98. doi:10
Möbius_strip
Concept in mathematics
The property of lying in Γ is local, i.e. if g : U → V is a homeomorphism between open sets of S and U is covered by open sets Ui with g restricted to
Pseudogroup
when restricted to U {\displaystyle U} then g 1 = g 2 {\displaystyle g_{1}=g_{2}} (this definition is inspired by the analytic continuation property of
(G,_X)-manifold
Problem in physics and celestial mechanics
gives a relative equilibrium motion in which the configuration remains an isometry of the initial configuration, as if the configuration was a rigid body
N-body_problem
Concept in topology
then be restricted to a bounded scalar sequence. If we further consider both spaces with the sup norm the extension map becomes an isometry. Indeed,
Stone–Čech_compactification
{Im} R}:\operatorname {Im} R\to \operatorname {Im} L} is a surjective isometry and L = U ∘ R . {\displaystyle L=U\circ R.} A linear map Λ : X → Y {\displaystyle
Inductive_tensor_product
Open problem on 3x+1 and x/2 functions
_{k=0}^{\infty }\left(T^{k}(x){\bmod {2}}\right)2^{k}.} The function Q is a 2-adic isometry. Consequently, every infinite parity sequence occurs for exactly one 2-adic
Collatz_conjecture
Universal property of metric spaces
spaces which aim at a subspace isometric to X, there is a unique (up to isometry) universal one, Aim(X), which in a sense of canonical isometric embeddings
Metric space aimed at its subspace
Metric_space_aimed_at_its_subspace
Matrix decomposition
axes. These directions happen to be mutually orthogonal. Apply first an isometry V ∗ {\displaystyle \mathbf {V} ^{*}} sending these directions to the
Singular_value_decomposition
Model of the extended complex plane plus a point at infinity
R 3 {\displaystyle \mathbf {R} ^{3}} (which, when restricted to the sphere, become the isometries of the sphere). In complex analysis, a meromorphic
Riemann_sphere
Function in mathematical analysis
τh defines a strongly continuous group of linear isometries of Lp. In the case p = ∞ the above property does not hold in general: actually, it exactly reduces
Modulus_of_continuity
Group of rotations in 3 dimensions
preserving the Euclidean distance between any two points (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations
3D_rotation_group
Branch of mathematics that studies the properties of groups
preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of X. If instead angles are preserved, one
Group_theory
Matrix representing a Euclidean rotation
of an isometry, a transformation that moves points without changing the distances between them. Rotations are distinguished from other isometries by two
Rotation_matrix
Differential operator in mathematics
which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions
Laplace_operator
Duality for locally compact abelian groups
} In particular, the Fourier transform is an L 2 {\displaystyle L^{2}} isometry from the complex-valued continuous functions of compact support on G {\displaystyle
Pontryagin_duality
Correspondence between quantum channels and quantum states
}(\rho )=\sum _{ij}\rho _{ij}C_{ij}=V^{\dagger }(\rho \otimes 1)V} , for an isometry V = ∑ i | i i ⟩ U i C i {\displaystyle V=\sum _{i}\vert ii\rangle U_{i}{\sqrt
Choi–Jamiołkowski_isomorphism
Theory of supersymmetry in four dimensions
unchanged. The first condition implies that the gauge symmetry belongs to the isometry group of the scalar manifold, while the second further restricts them to
4D_N_=_1_global_supersymmetry
Construct allowing differentiation of tangent vector fields of manifolds
following two properties: the connection is torsion-free, i.e., T∇ is zero, so that ∇XY − ∇YX = [X, Y]; parallel transport is an isometry, i.e., the inner
Affine_connection
248-dimensional exceptional simple Lie group
that the Jacobi identity is satisfied. The compact real form of E8 is the isometry group of the 128-dimensional exceptional compact Riemannian symmetric space
E8_(mathematics)
Set which cannot be assigned a meaningful "volume"
Such a measure is now called a Banach measure: it is invariant under all isometries, agrees with Lebesgue measure wherever Lebesgue measure is defined; and
Non-measurable_set
Stability conditions for triangulated cateogires
When restricted to the subset Stab ( D ) {\displaystyle \operatorname {Stab} ({\mathcal {D}})} of stability conditions that have the support property with
Bridgeland stability condition
Bridgeland_stability_condition
by E « F if E ~ F' ≤ F. In other words, E « F if there exists a partial isometry U ∈ M such that U*U = E and UU* ≤ F. For closed subspaces M and N where
Schröder–Bernstein theorems for operator algebras
Schröder–Bernstein_theorems_for_operator_algebras
Exterior algebraic map taking tensors from p forms to n-p forms
takes an orthonormal basis to an orthonormal basis. In this case, it is an isometry on the exterior algebra ⋀ V {\textstyle \bigwedge V} . The Hodge star is
Hodge_star_operator
Type of geometry
key projective invariant. The translations are described variously as isometries in metric space theory, as linear fractional transformations formally
Projective_geometry
Type of plane curve
John; Brooks, Jeff (1992), "A chord-stretching map of a convex loop is an isometry", Geometriae Dedicata, 41 (1): 51–62, doi:10.1007/BF00181542, MR 1147501
Convex_curve
Mathematical description of spacetime used in relativity
from the three spatial dimensions. In 3-dimensional Euclidean space, the isometry group (maps preserving the regular Euclidean distance) is the Euclidean
Minkowski_spacetime
Lie groups and their associated Lie algebras
their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the
Table_of_Lie_groups
Algebraic structure designed for geometry
all proper Euclidean isometries, which are always screw motions in 3-dimensional space, along with all improper Euclidean isometries, which includes reflections
Geometric_algebra
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
rotation in R 3 {\displaystyle \mathbb {R} ^{3}} : indeed it is clearly an isometry, since | q p q ∗ | 2 = q p q ∗ q p ∗ q ∗ = q p p ∗ q ∗ = | p | 2 {\displaystyle
Hopf_fibration
Analysis of datasets using techniques from topology
is Lipschitz continuous. Bottleneck distance is widely used in TDA. The isometry theorem asserts that the interleaving distance d I {\displaystyle d_{I}}
Topological_data_analysis
Real numbers with an added point at infinity
theory An extension does however exist in which all the algebraic properties, when restricted to defined operations in R ^ {\displaystyle {\widehat {\mathbb
Projectively extended real line
Projectively_extended_real_line
Theory of interwoven space and time by Albert Einstein
of this interval is a property of the general Lorentz transform (also called the Poincaré transformation), making it an isometry of spacetime. The general
Special_relativity
Theory of gravitation as curved spacetime
Gowdy, Robert H. (1974), "Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: Topologies and boundary conditions"
General_relativity
Graph drawing used to study Riemann surfaces
triangulation is a (cocompact) Fuchsian group representing a discrete set of isometries of the hyperbolic plane. In this case, the starting surface is the quotient
Dessin_d'enfant
Physical object which does not deform when forces or moments are exerted on it
non-zero translational motion) rigid body is E+(3), the subgroup of direct isometries of the Euclidean group in three dimensions (combinations of translations
Rigid_body
proportional fitting Iteratively reweighted least squares Itô calculus Itô isometry Itô's lemma Jaccard index Jackknife (statistics) Jackson network Jackson's
List_of_statistics_articles
(restricted) Sobolev space Hk 0(Ω) is defined as the closure of C∞ c(Ω) in the standard Sobolev space Hk(T2). H0 0(Ω) = L2(Ω). Vanishing properties on
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Graph representation in quantum mechanics
generated by a small set of spider tensors with phase parameters; within this restricted setting the diagrammatic rewrite rules can be made complete — complete
Tensor_network
Mathematical tool in quantum physics
{\displaystyle U} such that U † U = I {\displaystyle U^{\dagger }U=I} (a partial isometry), the ensemble { q i , | φ i ⟩ } {\displaystyle \{q_{i},|\varphi _{i}\rangle
Density_matrix
Poset representing certain properties of a polytope
the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces. If an abstract n-polytope is realized
Abstract_polytope
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
space L 2 ( Γ ∖ G , k ) {\displaystyle L^{2}(\Gamma \backslash G,k)} . The isometry is given by the map { ψ k : L 2 ( Γ ∖ H , k ) → L 2 ( Γ ∖ G , k ) ψ k (
Maass_wave_form
Property of a differential manifold that includes complex structures
under diffeomorphisms and also under shifts of the B-field, which are isometries of T ⊕ T ∗ {\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}} of the
Generalized_complex_structure
Representation of the symmetry group of spacetime in special relativity
open problem is the completion of the Bargmann–Wigner programme for the isometry group SO(D − 2, 1) of the de Sitter spacetime dSD−2. Ideally, the physical
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Representation theory of the symplectic group
{\displaystyle \left({\widehat {f}},{\widehat {g}}\right)=(f,g)} so defines an isometry of S {\displaystyle {\mathcal {S}}} onto itself. By density it extends
Oscillator_representation
Mathematical measure space associated to a random walk
geometric manner. For example, for groups of rank one (for example the isometry groups of hyperbolic spaces) the full Martin boundary is the same as the
Poisson_boundary
Polygon associated with a compact Riemann surface
ISBN 978-0-387-90465-8 Fenchel, Werner; Nielsen, Jakob (2003), Discontinuous groups of isometries in the hyperbolic plane, de Gruyter Studies in Mathematics, vol. 29, Walter
Fundamental_polygon
Coordinates system in an accelerating, "at rest" setting
ISBN 978-3-642-37276-6. Bel, L. (1994). "Born's group and Generalized isometries". In J. Diaz, M. Lorente (ed.). Relativity in General. Atlantica Séguier
Proper reference frame (flat spacetime)
Proper_reference_frame_(flat_spacetime)
Open convex self-dual cones
from the fact the D is complete for the Bergman metric, for which the isometries form a Lie group; by Montel's theorem, the group of biholomorphisms is
Symmetric_cone
be a Riemannian manifold and let G {\displaystyle G} be a Lie group of isometries of M {\displaystyle M} acting transitively on the sphere bundle S M .
Valuation_(geometry)
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RESTRICTED ISOMETRY-PROPERTY
RESTRICTED ISOMETRY-PROPERTY
RESTRICTED ISOMETRY-PROPERTY
RESTRICTED ISOMETRY-PROPERTY
RESTRICTED ISOMETRY-PROPERTY
RESTRICTED ISOMETRY-PROPERTY
RESTRICTED ISOMETRY-PROPERTY
RESTRICTED ISOMETRY-PROPERTY
RESTRICTED ISOMETRY-PROPERTY
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