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id stringlengths 40 40	title stringlengths 15 120	text stringlengths 41 3.14k	source stringclasses 2 values
03dbe8418a9a7c3f0c3d3d4a3347395f4aed3b8e	since z {\displaystyle z} is low-dimensional, the input	since z {\displaystyle z} is low-dimensional, the input can be easily transformed with z {\displaystyle z}, after that different linear learning methods to approximate the answer of the corresponding nonlinear kernel can be applied. there are different randomized feature maps to compute the approximations to the rbf ke...	wikipedia
577d84a55942e8990a93c4e92051c1eab0e91846	let x, x ′ ∈ r d {\displaystyle	let x, x ′ ∈ r d {\displaystyle \mathbf {x},\mathbf {x'} \in \mathbb {r} ^{d}} – samples of data, z: r d → r d {\displaystyle z:\mathbb {r} ^{d}\to \mathbb {r} ^{d}} – a randomized feature map (maps a single vector to a vector of higher dimensionality) so that the inner product between a pair of transformed points appr...	wikipedia
111d5dadd7c8927df8ffea475c65850c54f5b9b8	for a feature map Φ: x → f	for a feature map Φ: x → f {\displaystyle \phi :{\mathcal {x}}\to {\mathcal {f}}} with associated kernel k (x, x ′) = ⟨ Φ (x), Φ (x ′) ⟩ f {\displaystyle k(x,x')=\langle \phi (x),\phi (x')\rangle _{\mathcal {f}}}: equality k ^ = k ^ n, q k ^ q − 1 k ^ n, q t {\displaystyle {\hat {k}}={\hat {k}}_{n,q}{\hat {k}}_{q}^{-1}...	wikipedia
e011f243cb3e040ef8d88821d09e805f03cb2b21	for a feature map Φ: x → f	langle {\hat {\phi }}_{q}w\rangle _{i}=\langle \phi (x_{i}),w\rangle _{\mathcal {f}}}, i = 1, …, q {\displaystyle {\text{ }}i=1,\dots,q}, w ∈ f {\displaystyle w\in {\mathcal {f}}}. once again, a simple inspection shows that the feature map is only needed in the proof while the end result only depends on computing the k...	wikipedia
1083ec81ffa7d82d0d169f43c3a10b578c55b4df	defining k ^ n, q = x ^	defining k ^ n, q = x ^ x ^ q t {\displaystyle {\hat {k}}_{n,q}={\hat {x}}{\hat {x}}_{q}^{\text{t}}}, the proof is finished.	wikipedia
2aafefde87941e6e10629fbd59d9fceea69f4772	by the characterization for orthogonal matrix u ^	by the characterization for orthogonal matrix u ^ q {\displaystyle {\hat {u}}_{q}}: equality (u ^ q) t = (u ^ q) − 1 {\displaystyle ({\hat {u}}_{q})^{\text{t}}=({\hat {u}}_{q})^{-1}} holds. then, using the formula for the inverse of matrix multiplication (a b) − 1 = b − 1 a − 1 {\displaystyle (ab)^{-1}=b^{-1}a^{-1}} fo...	wikipedia
83687d9b479804eaba026048c1b6845dd75f8b36	however, defining k ~ = u ~ Σ	however, defining k ~ = u ~ Σ ^ q u ~ t {\displaystyle {\tilde {k}}={\tilde {u}}{\hat {\sigma }}_{q}{\tilde {u}}^{\text{t}}}, it can be computed the next way:	wikipedia
6bfb4b6f68b7181d93422031b8b151da65d314f3	u ~ = x ^ x ^ q	u ~ = x ^ x ^ q t u ^ q Σ ^ q − 1 {\displaystyle {\tilde {u}}={\hat {x}}{\hat {x}}_{q}^{\text{t}}{\hat {u}}_{q}{\hat {\sigma }}_{q}^{-1}} (u ~ {\displaystyle {\tilde {u}}} is not necessarily an orthogonal matrix).	wikipedia
5188e9ac7bcc143ae55ac14d0a13c826cb53de55	replacing v ^, Σ ^ {\displaystyle {\hat {v}},{\hat	replacing v ^, Σ ^ {\displaystyle {\hat {v}},{\hat {\sigma }}} by v ^ q {\displaystyle {\hat {v}}_{q}} and Σ ^ q {\displaystyle {\hat {\sigma }}_{q}}, an approximation for u ^ {\displaystyle {\hat {u}}} can be obtained:	wikipedia
e88db6c9d585028540bd9adb23503ba108ee0792	since u ^, v ^, u ^ q	since u ^, v ^, u ^ q and v ^ q {\displaystyle {\hat {u}},{\text{ }}{\hat {v}},{\hat {u}}_{q}{\text{ and }}{\hat {v}}_{q}} are orthogonal matrices,	wikipedia
62249aa66882c773efdecf5755a3ee08fe02635b	x ^ q = u ^ q Σ	x ^ q = u ^ q Σ ^ q 1 / 2 v ^ q t, k ^ q = u ^ q Σ ^ q u ^ q t, c ^ q = v ^ q Σ ^ q v ^ q t. {\displaystyle {\hat {x}}_{q}={\hat {u}}_{q}{\hat {\sigma }}_{q}^{1/2}{\hat {v}}_{q}^{\text{t}},{\text{ }}{\hat {k}}_{q}={\hat {u}}_{q}{\hat {\sigma }}_{q}{\hat {u}}_{q}^{t},{\text{ }}{\hat {c}}_{q}={\hat {v}}_{q}{\hat {\sigma ...	wikipedia
ef0ebf49b6b301f14444299212d3979181334a23	if u {\displaystyle u} and v {\displaystyle v}	if u {\displaystyle u} and v {\displaystyle v} are matrices with u {\displaystyle u} 's and v {\displaystyle v} 's in the columns and Σ {\displaystyle \sigma } is a diagonal p × m {\displaystyle p\times m} matrix having singular values σ i {\displaystyle \sigma _{i}} on the first k {\displaystyle k} -entries on the dia...	wikipedia
4ad2cce7b192e9e2c465feb08c7488337bb36853	applying singular-value decomposition (svd) to matrix a {\displaystyle	applying singular-value decomposition (svd) to matrix a {\displaystyle a} with dimensions p × m {\displaystyle p\times m} produces a singular system consisting of singular values { σ j } j = 1 k, (σ j > 0 ∀ j = 1, …, k), {\displaystyle \{\sigma _{j}\}_{j=1}^{k},{\text{ }}(\sigma _{j}>0{\text{ }}\forall j=1,\dots,k),} v...	wikipedia
f3ea0ed5ea84b1f52b4981bf307214bf0c432fd9	(k ^ n, q) i, j = k	(k ^ n, q) i, j = k (x i, x j), i = 1, …, n and j = 1, …, q. {\displaystyle ({\hat {k}}_{n,q})_{i,j}=k(x_{i},x_{j}),i=1,\dots,n{\text{ and }}j=1,\dots,q.}	wikipedia
85326c0aeabb9e4403e843bd4fecf7158483b0f2	(k ^ q) i, j = k (x	(k ^ q) i, j = k (x i, x j), i, j = 1, …, q {\displaystyle ({\hat {k}}_{q})_{i,j}=k(x_{i},x_{j}),i,j=1,\dots,q},	wikipedia
24e7df2a310383f07ab0f3d71d8f04d9e6d5ebbb	k ~ = k ^ n, q k	k ~ = k ^ n, q k ^ q − 1 k ^ n, q t {\displaystyle {\tilde {k}}={\hat {k}}_{n,q}{\hat {k}}_{q}^{-1}{\hat {k}}_{n,q}^{\text{t}}}, where	wikipedia
c14682c8d3c841f2133c3f063d5ad76623fd4ee9	k ^ {\displaystyle {\hat {k}}} is a kernel	k ^ {\displaystyle {\hat {k}}} is a kernel matrix for some kernel method. consider the first q < n {\displaystyle q<n} points in the training set. then there exists a matrix k ~ {\displaystyle {\tilde {k}}} of rank q {\displaystyle q}:	wikipedia
cbbb9092f868c0a85b574f7faff1f4ee83158d19	if n {\displaystyle n} is the number of	if n {\displaystyle n} is the number of training examples, the storage and computational cost required to find the solution of the problem using general kernel method is o (n 2) {\displaystyle o(n^{2})} and o (n 3) {\displaystyle o(n^{3})} respectively. the nyström approximation can allow a significant speed-up of the ...	wikipedia
a3d2cec0c2e420e8fb8e2aaa9d65c1ea0e376a6a	kernel methods (for instance, support vector machines or	kernel methods (for instance, support vector machines or gaussian processes) project data points into a high-dimensional or infinite-dimensional feature space and find the optimal splitting hyperplane. in the kernel method the data is represented in a kernel matrix (or, gram matrix). many algorithms can solve machine l...	wikipedia
d9af3cdf6251d24968a90442b7061ee996b29cab	julieta castellanos is a honduran sociologist and the	julieta castellanos is a honduran sociologist and the dean of the national autonomous university of honduras (unah) since 2009. castellanos is known for campaigning against violence in honduras, focusing on both drug cartels and police corruption. she has advocated for both judicial and police reform. castellanos found...	wikipedia
942a7a3fa78368a7e5c62460cd37e45753011f36	castellanos was presented with the inaugural martin luther	castellanos was presented with the inaugural martin luther king, jr. award on 20 april 2012 at a ceremony sponsored by the martin luther king, jr. foundation and the instituto hondureño de cultura interamericana.	wikipedia
d4d3dfdd0c257b025c657cb6e197382870e73baf	in october 2011, castellanos' 22-year-old son was kidnapped	in october 2011, castellanos' 22-year-old son was kidnapped and murdered by honduran national police. the incident called attention to the degree of corruption within the honduran police. castellanos called for an end to foreign aid for the honduran police and military, demanding that they "stop feeding the beast."	wikipedia
866ddb032da2644e4e324ac264adf86859ea1ea7	castellanos has been a vocal advocate for police	castellanos has been a vocal advocate for police reform and measures to curb violent crime in honduras. she pushed for an international commission to oversee a purge of the police, an idea that was approved by the national congress. castellanos has also weighed in on gun politics in honduras, calling on the honduran ar...	wikipedia
90635b4993b94f45c1014c65230927e5f926bfd4	shortly after her appointment, president manuel zelaya was	shortly after her appointment, president manuel zelaya was ousted in a coup. during a demonstration by university students, castellanos was pushed down by police when she attempted to stop them from breaking up the gathering. during her tenure as rector, she has clashed with sitraunah, the union for the university's em...	wikipedia
23cbf347e68edf590891d5be43a9104bb505adcc	castellanos was elected to a four-year term as	castellanos was elected to a four-year term as rector of the national autonomous university of honduras in 2009. at the time of her appointment she served as coordinator of the unah violence observatory and director of the instituto universitario en democracia paz y seguridad (institute for democracy, peace and securit...	wikipedia
9897de04df035df54d48c1e0f4673727c71119bb	after completing her studies castellanos became a professor	after completing her studies castellanos became a professor at unah in 1978. she was head of the social sciences department and president of the association of teachers from 1997 to 2001. castellanos also served as president of the association of professors of the university center of general studies (cueg) in 1986, co...	wikipedia
c6f3a55c7a49da5a4acf37e293cf33bee2c7e997	julieta castellanos ruiz was born in san francisco	julieta castellanos ruiz was born in san francisco de becerra, olancho on 8 january 1954 to rafael castellanos of santa bárbara, and ernestina ruiz of olancho. she grew up in the rural honduran sugar fields. in 1968 her father brought her an examination for admission to the normal school for girls in the city of teguci...	wikipedia
4864b0aed295a6091273671d48ea70cd88ca0022	julieta castellanos (born 8 january 1954) is a	julieta castellanos (born 8 january 1954) is a honduran sociologist and the dean of the national autonomous university of honduras (unah) since 2009. castellanos is known for campaigning against violence in honduras, focusing on both drug cartels and police corruption. she has advocated for both judicial and police ref...	wikipedia
11203b8e414f23ab9a5fe7646b5314d618609443	the first article in the "research problems" section	the first article in the "research problems" section of the american mathematical monthly was dedicated by victor klee to hadwiger, on the occasion of his 60th birthday, in honor of hadwiger's work editing a column on unsolved problems in the journal elemente der mathematik.	wikipedia
fc92680d5326d5101e54c38e9643ccabb41dac1c	hadwiger was one of the principal developers of	hadwiger was one of the principal developers of a swiss rotor machine for encrypting military communications, known as nema. the swiss, fearing that the germans and allies could read messages transmitted on their enigma cipher machines, enhanced the system by using ten rotors instead of five. the system was used by the...	wikipedia
c5422ff1f6d79eaa1c425f70513e56b17ad2c7c0	hadwiger proved a theorem characterizing eutactic stars, systems	hadwiger proved a theorem characterizing eutactic stars, systems of points in euclidean space formed by orthogonal projection of higher-dimensional cross polytopes. he found a higher-dimensional generalization of the space-filling hill tetrahedra. and his 1957 book vorlesungen über inhalt, oberfläche und isoperimetrie ...	wikipedia
4ac6a79f6394c68f106af4cbce946e11eb98a24f	the hadwiger–finsler inequality, proven by hadwiger with paul	the hadwiger–finsler inequality, proven by hadwiger with paul finsler, is an inequality relating the side lengths and area of any triangle in the euclidean plane. it generalizes weitzenböck's inequality and was generalized in turn by pedoe's inequality. in the same 1937 paper in which hadwiger and finsler published thi...	wikipedia
902f52f6554ada59abef2a61c43e0774ab4c1653	hadwiger's theorem in integral geometry classifies the isometry-invariant	hadwiger's theorem in integral geometry classifies the isometry-invariant valuations on compact convex sets in d -dimensional euclidean space. according to this theorem, any such valuation can be expressed as a linear combination of the intrinsic volumes; for instance, in two dimensions, the intrinsic volumes are the a...	wikipedia
c661c03c361e96c2dae11b66ff35a239d366ebc3	although born in karlsruhe, germany, hadwiger grew up	although born in karlsruhe, germany, hadwiger grew up in bern, switzerland. he did his undergraduate studies at the university of bern, where he majored in mathematics but also studied physics and actuarial science. he continued at bern for his graduate studies, and received his ph.d. in 1936 under the supervision of w...	wikipedia
e41dba2bf3b262806b76f653c85b23df3dfb56b0	hugo hadwiger (23 december 1908 in karlsruhe, germany	hugo hadwiger (23 december 1908 in karlsruhe, germany – 29 october 1981 in bern, switzerland) was a swiss mathematician, known for his work in geometry, combinatorics, and cryptography.	wikipedia
0f80d48275f7373a2106f594ac935d493d9328ce	hinuma station (涸沼駅, hinuma-eki) is a passenger railway	hinuma station (涸沼駅, hinuma-eki) is a passenger railway station in the town of hokota, ibaraki prefecture, japan operated by the third sector kashima rinkai railway.	wikipedia
9407175adfd37e1ef88924852678fd94492a16e3	maurice krishaber, born krishaber mór in feketehegy in	maurice krishaber, born krishaber mór in feketehegy in vojvodina on 3 april 1836 and died in mulhouse (france) on 10 april 1883, was a naturalised french hungarian otorhinolaryngologist.	wikipedia
6cf1daa58881721ac4024bcbc0ac0afef93e91b2	in 1875, together with jules ladreit de la	in 1875, together with jules ladreit de la charrière and Émile isambert, he founded the annales des maladies de l'oreille et du larynx (otoscopie, laryngoscopie, rhinoscopie).	wikipedia
4ddc249f21e493e47d93b7ca9202f6faf6204866	in 1874, Émile isambert, son of françois-andré isambert,	in 1874, Émile isambert, son of françois-andré isambert, opened the first laryngology clinic at lariboisière hospital and at his death in 1876, he was succeeded by maurice raynaud then by adrien proust, the writer's father, who was especially interested in hygiene and the fight against cholera, the laryngology room bei...	wikipedia
590a899a5667918e8d2594d3d398d6b579222fbb	maurice krishaber, "under the category cerebro-cardiac neuropathy (névropathie	maurice krishaber, "under the category cerebro-cardiac neuropathy (névropathie cérébro-cardiaque) reported 38 patients showing a mixture of anxiety, fatigue and depression. over one-third of these patients complained of baffing and unpleasant mental experiences, consisting of the loss of a feeling of reality. krishaber...	wikipedia