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In a prepare-and-measure scenario, a universal quantum computer prepares a thermal state, which is then sampled by measurements. This can reduce the time required to train a deep restricted Boltzmann machine, and provide a richer and more comprehensive framework for deep learning than classical computing. The same quan... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum sampling techniques | 176 | 987 | null |
Section: Machine learning with quantum computers > Quantum Convolution Neural Network. A novel design for multi-dimensional vectors that uses circuits as convolution filters is QCNN. It was inspired by the advantages of CNNs and the power of QML. It is made using a combination of a variational quantum circuit(VQC) and ... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum Convolution Neural Network | 334 | 1,597 | null |
Quantum neural networks take advantage of the hierarchical structures, and for each subsequent layer, the number of qubits from the preceding layer is decreased by a factor of two. For n input qubits, these structure have O(log(n)) layers, allowing for shallow circuit depth. Additionally, they are able to avoid "barren... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum Convolution Neural Network | 290 | 1,464 | null |
Section: Machine learning with quantum computers > Fully quantum machine learning. In the most general case of quantum machine learning, both the learning device and the system under study, as well as their interaction, are fully quantum. This section gives a few examples of results on this topic. One class of problem ... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Fully quantum machine learning | 337 | 1,870 | null |
Section: Machine learning with quantum computers > Explainable quantum machine learning. The need for models that can be understood by humans emerges in quantum machine learning in analogy to classical machine learning and drives the research field of explainable quantum machine learning (or XQML in analogy to XAI/XML)... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Explainable quantum machine learning | 213 | 1,078 | null |
Section: Quantum learning theory. Quantum learning theory pursues a mathematical analysis of the quantum generalizations of classical learning models and of the possible speed-ups or other improvements that they may provide. The framework is very similar to that of classical computational learning theory, but the learn... | Wikipedia - Quantum machine learning - Quantum learning theory | 342 | 1,707 | null |
In the model of quantum exact learning, the learner can make membership queries in quantum superposition. If the complexity of the learner is measured by the number of membership queries it makes, then quantum exact learners can be polynomially more efficient than classical learners for some concept classes, but not mo... | Wikipedia - Quantum machine learning - Quantum learning theory | 346 | 1,579 | null |
However, for learning under some fixed distribution D, quantum examples can be very helpful, for example for learning DNF under the uniform distribution. When considering time complexity, there exist concept classes that can be PAC-learned efficiently by quantum learners, even from classical examples, but not by classi... | Wikipedia - Quantum machine learning - Quantum learning theory | 186 | 994 | null |
Section: Implementations and experiments. The earliest experiments were conducted using the adiabatic D-Wave quantum computer, for instance, to detect cars in digital images using regularized boosting with a nonconvex objective function in a demonstration in 2009. Many experiments followed on the same architecture, and... | Wikipedia - Quantum machine learning - Implementations and experiments | 330 | 1,793 | null |
Once the vectors are defined on the feature space, the quantum support vector machine was implemented to classify the unknown input vector. The readout avoids costly quantum tomography by reading out the final state in terms of direction (up/down) of the NMR signal. Photonic implementations are attracting more attentio... | Wikipedia - Quantum machine learning - Implementations and experiments | 340 | 1,743 | null |
Since 2016, IBM has launched an online cloud-based platform for quantum software developers, called the IBM Q Experience. This platform consists of several fully operational quantum processors accessible via the IBM Web API. In doing so, the company is encouraging software developers to pursue new algorithms through a ... | Wikipedia - Quantum machine learning - Implementations and experiments | 327 | 1,788 | null |
Section: Skepticism. While machine learning itself is now not only a research field but an economically significant and fast growing industry and quantum computing is a well established field of both theoretical and experimental research, quantum machine learning remains a purely theoretical field of studies. Attempts ... | Wikipedia - Quantum machine learning - Skepticism | 342 | 1,807 | null |
Section: Hardware. Display: A 2.88-inch touchscreen for interactive user input. Input: push-to-talk button to activate voice commands; scroll wheel; Gyroscope; Magnetometer; Accelerometer; GPS. Camera: 8 MP single camera, with a resolution of 3264x2448, allowing for the connected external AI to use computer vision. Aud... | Wikipedia - Rabbit r1 - Hardware | 165 | 719 | null |
Section: Software. The Rabbit r1 runs on Rabbit OS, based on the Android Open Source Project (AOSP), specifically version 13. Lyu has claimed that Rabbit OS runs with a "very bespoke AOSP." The device employs a large action model (LAM) designed to perform actions and assist with tasks like web searches, streaming music... | Wikipedia - Rabbit r1 - Software | 207 | 1,001 | null |
Section: Reception > Reviews. The r1 was met with strong criticism immediately after Rabbit began shipping the device. Some reviews questioned what the device was able to do that a smartphone could not, while comparing it to the similar Humane Ai Pin. YouTuber Marques Brownlee called the device "barely reviewable". And... | Wikipedia - Rabbit r1 - Reception > Reviews | 308 | 1,446 | null |
Section: Controversies > GAMA project. Rabbit Inc has garnered attention due to allegations surrounding its funding and the company's past projects. The company came under scrutiny when Stephen Findeisen, known as Coffeezilla on YouTube, published a video in May 2024, alleging that Rabbit Incorporation was "built on a ... | Wikipedia - Rabbit r1 - Controversies > GAMA project | 339 | 1,625 | null |
Section: Controversies > Security. In June 2024, Engadget reported that the Rabbitude team, a community reverse engineering project, had gained access to the r1's codebase revealing that r1's software contained several hardcoded API keys in its code for ElevenLabs, Microsoft Azure, Yelp, and Google Maps, potentially al... | Wikipedia - Rabbit r1 - Controversies > Security | 236 | 1,120 | null |
Section: Definitions > Rademacher complexity of a set. Given a set A β R m {\displaystyle A\subseteq \mathbb {R} ^{m}} , the Rademacher complexity of A is defined as follows:: 326 Rad β‘ ( A ) := 1 m E Ο [ sup a β A β i = 1 m Ο i a i ] {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\su... | Wikipedia - Rademacher complexity - Definitions > Rademacher complexity of a set | 336 | 759 | null |
Pr ( Ο i = + 1 ) = Pr ( Ο i = β 1 ) = 1 / 2 {\displaystyle \Pr(\sigma _{i}=+1)=\Pr(\sigma _{i}=-1)=1/2} for i = 1 , 2 , β¦ , m {\displaystyle i=1,2,\dots ,m} , and a = ( a 1 , β¦ , a m ) {\displaystyle a=(a_{1},\ldots ,a_{m})} . Some authors take the absolute value of the sum before taking the supremum, but if A {\displa... | Wikipedia - Rademacher complexity - Definitions > Rademacher complexity of a set | 152 | 368 | null |
Then, the empirical Rademacher complexity of F {\displaystyle {\mathcal {F}}} given S {\displaystyle S} is defined as: Rad S β‘ ( F ) = 1 m E Ο [ sup f β F | β i = 1 m Ο i f ( z i ) | ] {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{f\in {\mathcal {F}}}\left|\sum... | Wikipedia - Rademacher complexity - Definitions > Rademacher complexity of a function class | 350 | 773 | null |
{\mathcal {F}}\}} The worst case empirical Rademacher complexity is Rad Β― m ( F ) = sup S = { z 1 , β¦ , z m } Rad S β‘ ( F ) {\displaystyle {\overline {\operatorname {Rad} }}_{m}({\mathcal {F}})=\sup _{S=\{z_{1},\dots ,z_{m}\}}\operatorname {Rad} _{S}({\mathcal {F}})} Let P {\displaystyle P} be a probability distributio... | Wikipedia - Rademacher complexity - Definitions > Rademacher complexity of a function class | 150 | 348 | null |
{\mathcal {F}}\}} The worst case empirical Rademacher complexity is Rad Β― m ( F ) = sup S = { z 1 , β¦ , z m } Rad S β‘ ( F ) {\displaystyle {\overline {\operatorname {Rad} }}_{m}({\mathcal {F}})=\sup _{S=\{z_{1},\dots ,z_{m}\}}\operatorname {Rad} _{S}({\mathcal {F}})} Let P {\displaystyle P} be a probability distributio... | Wikipedia - Rademacher complexity - Definitions > Rademacher complexity of a function class | 368 | 911 | null |
The Rademacher complexity of a set Rad β‘ ( A ) := 1 m E Ο [ sup a β A β i = 1 m Ο i a i ] {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]} can be rewritten as Rad β‘ ( A ) := 1 m 2 m β Ο β { β 1 / m , + 1 / m } m [ sup a β A β¨ Ο , a β©... | Wikipedia - Rademacher complexity - Intuition | 315 | 713 | null |
The directions are along the vertices of a hypercube. Thus, we can also write it as Rad β‘ ( A ) := 1 2 m 1 2 m β 1 β Ο β { β 1 / m , + 1 / m } m / { β 1 , + 1 } [ sup a β A β¨ Ο , a β© β inf a β A β¨ Ο , a β© ] {\displaystyle \operatorname {Rad} (A):={\frac {1}{2{\sqrt {m}}}}{\frac {1}{2^{m-1}}}\sum _{\sigma \in \{-1/{\sqr... | Wikipedia - Rademacher complexity - Intuition | 323 | 678 | null |
Thus, we can also write it as Rad β‘ ( A ) := 1 2 m 1 2 m β 1 β Ο β { β 1 / m , + 1 / m } m / { β 1 , + 1 } [ sup a β A β¨ Ο , a β© β inf a β A β¨ Ο , a β© ] {\displaystyle \operatorname {Rad} (A):={\frac {1}{2{\sqrt {m}}}}{\frac {1}{2^{m-1}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}}\left[\sup _{a\i... | Wikipedia - Rademacher complexity - Intuition | 373 | 827 | null |
Section: Examples. Singleton sets have 0 width in any direction, so it has Rademacher complexity 0.: 56 If A = { ( 1 , 1 ) , ( 1 , 2 ) } β R 2 {\displaystyle A=\{(1,1),(1,2)\}\subset \mathbb {R} ^{2}} , then it has average width 1 / 2 {\displaystyle 1/{\sqrt {2}}} along the two diagonal directions of the square, so it ... | Wikipedia - Rademacher complexity - Examples | 300 | 830 | null |
Section: Using the Rademacher complexity > Bounding the representativeness. In machine learning, it is desired to have a training set that represents the true distribution of some sample data S {\displaystyle S} . This can be quantified using the notion of representativeness. Denote by P {\displaystyle P} the probabili... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Bounding the representativeness | 308 | 1,201 | null |
We omit the index and write f {\displaystyle f} instead of f h {\displaystyle f_{h}} when the underlying hypothesis is irrelevant. Define: L P ( f ) := E z βΌ P [ f ( z ) ] {\displaystyle L_{P}(f):=\mathbb {E} _{z\sim P}[f(z)]} β the expected error of some error function f β F {\displaystyle f\in {\mathcal {F}}} on the ... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Bounding the representativeness | 225 | 580 | null |
Define: L P ( f ) := E z βΌ P [ f ( z ) ] {\displaystyle L_{P}(f):=\mathbb {E} _{z\sim P}[f(z)]} β the expected error of some error function f β F {\displaystyle f\in {\mathcal {F}}} on the real distribution P {\displaystyle P} ; L S ( f ) := 1 m β i = 1 m f ( z i ) {\displaystyle L_{S}(f):={1 \over m}\sum _{i=1}^{m}f(z... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Bounding the representativeness | 376 | 1,027 | null |
The representativeness of the sample S {\displaystyle S} , with respect to P {\displaystyle P} and F {\displaystyle {\mathcal {F}}} , is defined as: Rep P β‘ ( F , S ) := sup f β F ( L P ( f ) β L S ( f ) ) {\displaystyle \operatorname {Rep} _{P}({\mathcal {F}},S):=\sup _{f\in F}(L_{P}(f)-L_{S}(f))} Smaller representati... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Bounding the representativeness | 206 | 696 | null |
The expected representativeness of a sample can be bounded above by the Rademacher complexity of the function class: If F {\displaystyle {\mathcal {F}}} is a set of functions with range within [ 0 , 1 ] {\displaystyle [0,1]} , then: 326 Rad P , m β‘ ( F ) β ln β‘ 2 2 m β€ E S βΌ P m [ Rep P β‘ ( F , S ) ] β€ 2 Rad P , m β‘ ( ... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Bounding the representativeness | 346 | 868 | null |
When the error function is binary (0-1 loss), for every Ξ΄ > 0 {\displaystyle \delta >0} , sup f β F ( L P ( f ) β L S ( f ) ) β€ 2 Rad S β‘ ( F ) + 4 2 ln β‘ ( 4 / Ξ΄ ) m {\displaystyle \sup _{f\in {\mathcal {F}}}(L_{P}(f)-L_{S}(f))\leq 2\operatorname {Rad} _{S}({\mathcal {F}})+4{\sqrt {2\ln(4/\delta ) \over m}}} with prob... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Bounding the generalization error | 350 | 828 | null |
Section: Using the Rademacher complexity > Oracle inequalities. Let the Bayes risk L β = inf f L P ( f ) {\displaystyle L^{*}=\inf _{f}L_{P}(f)} , where f {\displaystyle f} can be any measurable function. Let the function class F {\displaystyle {\mathcal {F}}} be split into "complexity classes" F r {\displaystyle {\mat... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Oracle inequalities | 291 | 770 | null |
For any dataset S {\displaystyle S} , let f ^ {\displaystyle {\hat {f}}} be a minimizer of L S ( f ) + p ( f ) {\displaystyle L_{S}(f)+p(f)} . If sup f β F r | L P ( f ) β L S ( f ) | β€ p r , β r {\displaystyle \sup _{f\in {\mathcal {F}}_{r}}|L_{P}(f)-L_{S}(f)|\leq p_{r},\quad \forall r} then we have the oracle inequal... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Oracle inequalities | 319 | 609 | null |
If we further assume r β€ s implies F r β F s and p r β€ p s {\displaystyle r\leq s{\text{ implies }}{\mathcal {F}}_{r}\subseteq {\mathcal {F}}_{s}{\text{ and }}p_{r}\leq p_{s}} and β r , sup f β F r ( L P ( f ) β L P ( f r β ) β 2 ( L S ( f ) β L S ( f r β ) ) ) β€ 2 p r / 7 sup f β F r ( L S ( f ) β L S ( f r β ) β 2 ( ... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Oracle inequalities | 310 | 571 | null |
{F}}_{r}}\left(L_{S}(f)-L_{S}\left(f_{r}^{*}\right)-2\left(L_{P}(f)-L_{P}\left(f_{r}^{*}\right)\right)\right)&\leq 2p_{r}/7\end{aligned}}} then we have the oracle inequality L P ( f ^ ) β L β β€ inf r ( inf f β F r L P ( f ) β L β + 3 p r ) {\displaystyle L_{P}({\widehat {f}})-L^{*}\leq \inf _{r}\left(\inf _{f\in {\math... | Wikipedia - Rademacher complexity - Using the Rademacher complexity > Oracle inequalities | 228 | 376 | null |
Section: Bounding the Rademacher complexity. Since smaller Rademacher complexity is better, it is useful to have upper bounds on the Rademacher complexity of various function sets. The following rules can be used to upper-bound the Rademacher complexity of a set A β R m {\displaystyle A\subset \mathbb {R} ^{m}} .: 329β... | Wikipedia - Rademacher complexity - Bounding the Rademacher complexity | 340 | 1,087 | null |
In particular, if all vectors in A {\displaystyle A} are operated by a contraction mapping, then Rad(A) strictly decreases. The Rademacher complexity of the convex hull of A {\displaystyle A} equals Rad(A). (Massart Lemma) The Rademacher complexity of a finite set grows logarithmically with the set size. Formally, let ... | Wikipedia - Rademacher complexity - Bounding the Rademacher complexity | 318 | 894 | null |
Section: Bounding the Rademacher complexity > Bounds related to the VC dimension. Let H {\displaystyle H} be a set family whose VC dimension is d {\displaystyle d} . It is known that the growth function of H {\displaystyle H} is bounded as: for all m > d + 1 {\displaystyle m>d+1} : Growth β‘ ( H , m ) β€ ( e m / d ) d {\... | Wikipedia - Rademacher complexity - Bounding the Rademacher complexity > Bounds related to the VC dimension | 228 | 674 | null |
The set-family H β© h {\displaystyle H\cap h} can be considered as a set of binary vectors over R m {\displaystyle \mathbb {R} ^{m}} . Substituting this in Massart's lemma gives: Rad β‘ ( H β© h ) β€ 2 d log β‘ ( e m / d ) m {\displaystyle \operatorname {Rad} (H\cap h)\leq {\sqrt {2d\log(em/d) \over m}}} With more advanced ... | Wikipedia - Rademacher complexity - Bounding the Rademacher complexity > Bounds related to the VC dimension | 230 | 684 | null |
Section: Bounding the Rademacher complexity > Bounds related to linear classes. The following bounds are related to linear operations on S {\displaystyle S} β a constant set of m {\displaystyle m} vectors in R n {\displaystyle \mathbb {R} ^{n}} .: 332β333 Define A 2 = { ( w β
x 1 , β¦ , w β
x m ) β£ β w β 2 β€ 1 } = {\dis... | Wikipedia - Rademacher complexity - Bounding the Rademacher complexity > Bounds related to linear classes | 176 | 490 | null |
The following bounds are related to linear operations on S {\displaystyle S} β a constant set of m {\displaystyle m} vectors in R n {\displaystyle \mathbb {R} ^{n}} .: 332β333 Define A 2 = { ( w β
x 1 , β¦ , w β
x m ) β£ β w β 2 β€ 1 } = {\displaystyle A_{2}=\{(w\cdot x_{1},\ldots ,w\cdot x_{m})\mid \|w\|_{2}\leq 1\}=} th... | Wikipedia - Rademacher complexity - Bounding the Rademacher complexity > Bounds related to linear classes | 341 | 790 | null |
Then: Rad β‘ ( A 2 ) β€ max i β x i β 2 m {\displaystyle \operatorname {Rad} (A_{2})\leq {\max _{i}\|x_{i}\|_{2} \over {\sqrt {m}}}} Define A 1 = { ( w β
x 1 , β¦ , w β
x m ) β£ β w β 1 β€ 1 } = {\displaystyle A_{1}=\{(w\cdot x_{1},\ldots ,w\cdot x_{m})\mid \|w\|_{1}\leq 1\}=} the set of dot-products of the vectors in S {\d... | Wikipedia - Rademacher complexity - Bounding the Rademacher complexity > Bounds related to linear classes | 270 | 547 | null |
Then, for every integer M > 0 {\displaystyle M>0} : Rad β‘ ( A ) β€ c β
2 β M m + 6 c m β
β i = 1 M 2 β i log β‘ ( N c β
2 β i ext ( A ) ) {\displaystyle \operatorname {Rad} (A)\leq {c\cdot 2^{-M} \over {\sqrt {m}}}+{6c \over m}\cdot \sum _{i=1}^{M}2^{-i}{\sqrt {\log \left(N_{c\cdot 2^{-i}}^{\text{ext}}(A)\right)}}} In pa... | Wikipedia - Rademacher complexity - Bounding the Rademacher complexity > Bounds related to covering numbers | 350 | 774 | null |
Section: Gaussian complexity. Gaussian complexity is a similar complexity with similar physical meanings, and can be obtained from the Rademacher complexity using the random variables g i {\displaystyle g_{i}} instead of Ο i {\displaystyle \sigma _{i}} , where g i {\displaystyle g_{i}} are Gaussian i.i.d. random variab... | Wikipedia - Rademacher complexity - Gaussian complexity | 158 | 516 | null |
Section: Gaussian complexity > Equivalence of Rademacher and Gaussian complexity. Given a set A β R n {\displaystyle A\subseteq \mathbb {R} ^{n}} then it holds that: G ( A ) 2 log β‘ n β€ Rad ( A ) β€ Ο 2 G ( A ) {\displaystyle {\frac {G(A)}{2{\sqrt {\log {n}}}}}\leq {\text{Rad}}(A)\leq {\sqrt {\frac {\pi }{2}}}G(A)} Wher... | Wikipedia - Rademacher complexity - Gaussian complexity > Equivalence of Rademacher and Gaussian complexity | 250 | 708 | null |
Section: Mathematics > Kernel method. Given a feature map Ο : R d β V {\textstyle \phi :\mathbb {R} ^{d}\to V} , where V {\textstyle V} is a Hilbert space (more specifically, a reproducing kernel Hilbert space), the kernel trick replaces inner products in feature space β¨ Ο ( x i ) , Ο ( x j ) β© V {\displaystyle \langle... | Wikipedia - Random feature - Mathematics > Kernel method | 284 | 750 | null |
Section: Mathematics > Random kernel method. The problem with kernel methods is that the kernel matrix K X {\textstyle K_{X}} has size N Γ N {\textstyle N\times N} . This becomes computationally infeasible when N {\textstyle N} reaches the order of a million. The random kernel method replaces the kernel function k {\te... | Wikipedia - Random feature - Mathematics > Random kernel method | 232 | 759 | null |
This converts kernel linear regression into linear regression in feature space, kernel SVM into SVM in feature space, etc. Since we have K X β Z X T Z X {\displaystyle K_{X}\approx Z_{X}^{T}Z_{X}} where Z X = [ z ( x 1 ) , β¦ , z ( x N ) ] {\displaystyle Z_{X}=[z(x_{1}),\dots ,z(x_{N})]} , these methods no longer involv... | Wikipedia - Random feature - Mathematics > Random kernel method | 164 | 444 | null |
Section: Random Fourier feature > Radial basis function kernel. The radial basis function (RBF) kernel on two samples x i , x j β R d {\displaystyle x_{i},x_{j}\in \mathbb {R} ^{d}} is defined as k ( x i , x j ) = exp β‘ ( β β x i β x j β 2 2 Ο 2 ) {\displaystyle k(x_{i},x_{j})=\exp \left(-{\frac {\|x_{i}-x_{j}\|^{2}}{2... | Wikipedia - Random feature - Random Fourier feature > Radial basis function kernel | 210 | 517 | null |
The radial basis function (RBF) kernel on two samples x i , x j β R d {\displaystyle x_{i},x_{j}\in \mathbb {R} ^{d}} is defined as k ( x i , x j ) = exp β‘ ( β β x i β x j β 2 2 Ο 2 ) {\displaystyle k(x_{i},x_{j})=\exp \left(-{\frac {\|x_{i}-x_{j}\|^{2}}{2\sigma ^{2}}}\right)} where β x i β x j β 2 {\displaystyle \|x_{... | Wikipedia - Random feature - Random Fourier feature > Radial basis function kernel | 405 | 897 | null |
It can be approximated by a random Fourier feature map z : R d β R 2 D {\displaystyle z:\mathbb {R} ^{d}\to \mathbb {R} ^{2D}} : z ( x ) := 1 D [ cos β‘ β¨ Ο 1 , x β© , sin β‘ β¨ Ο 1 , x β© , β¦ , cos β‘ β¨ Ο D , x β© , sin β‘ β¨ Ο D , x β© ] T {\displaystyle z(x):={\frac {1}{\sqrt {D}}}[\cos \langle \omega _{1},x\rangle ,\sin \lan... | Wikipedia - Random feature - Random Fourier feature > Radial basis function kernel | 310 | 749 | null |
Section: Random Fourier feature > Random Fourier features. By Bochner's theorem, the above construction can be generalized to arbitrary positive definite shift-invariant kernel k ( x , y ) = k ( x β y ) {\displaystyle k(x,y)=k(x-y)} . Define its Fourier transform p ( Ο ) = 1 2 Ο β« R d e β j β¨ Ο , Ξ β© k ( Ξ ) d Ξ {\disp... | Wikipedia - Random feature - Random Fourier feature > Random Fourier features | 225 | 680 | null |
Section: Random Fourier feature > Neural network interpretation. Given a random Fourier feature map z {\displaystyle z} , training the feature on a dataset by featurized linear regression is equivalent to fitting complex parameters ΞΈ 1 , β¦ , ΞΈ D β C {\displaystyle \theta _{1},\dots ,\theta _{D}\in \mathbb {C} } such th... | Wikipedia - Random feature - Random Fourier feature > Neural network interpretation | 208 | 658 | null |
Given a random Fourier feature map z {\displaystyle z} , training the feature on a dataset by featurized linear regression is equivalent to fitting complex parameters ΞΈ 1 , β¦ , ΞΈ D β C {\displaystyle \theta _{1},\dots ,\theta _{D}\in \mathbb {C} } such that f ΞΈ ( x ) = R e ( β k ΞΈ k e i β¨ Ο k , x β© ) {\displaystyle f_{... | Wikipedia - Random feature - Random Fourier feature > Neural network interpretation | 521 | 1,400 | null |
Section: Other examples > Random binning features. A random binning features map partitions the input space using randomly shifted grids at randomly chosen resolutions and assigns to an input point a binary bit string that corresponds to the bins in which it falls. The grids are constructed so that the probability that... | Wikipedia - Random feature - Other examples > Random binning features | 244 | 930 | null |
Section: Historical context. In NIPS 2006, deep learning had just become competitive with linear models like PCA and linear SVMs for large datasets, and people speculated about whether it could compete with kernel SVMs. However, there was no way to train kernel SVM on large datasets. The two authors developed the rando... | Wikipedia - Random feature - Historical context | 177 | 726 | null |
Section: Background. RHML emerged in the context of the rise of big data analytics and artificial intelligence for intelligent tasks like sense-making and decision-making. As machine learning advanced to take on more roles, researchers realized fully autonomous systems had limitations and needed human guidance. RHML ex... | Wikipedia - Reciprocal human machine learning - Background | 168 | 901 | null |
In more detail, we have to statistically estimate: β Ο L ( Ο ) = β Ο β« d z q Ο ( z ) f ( z ) {\displaystyle \nabla _{\phi }L(\phi )=\nabla _{\phi }\int dz\;q_{\phi }(z)f(z)} The REINFORCE estimator, widely used in reinforcement learning and especially policy gradient, uses the following equality: β Ο L ( Ο ) = β« d z q ... | Wikipedia - Reparameterization trick - Mathematics > REINFORCE estimator | 349 | 770 | null |
This gives: L ( Ο ) = E Ο΅ βΌ p ( Ο΅ ) [ f ( g Ο ( Ο΅ ) ) ] {\displaystyle L(\phi )=\mathbb {E} _{\epsilon \sim p(\epsilon )}[f(g_{\phi }(\epsilon ))]} Now, the gradient can be estimated as: β Ο L ( Ο ) = E Ο΅ βΌ p ( Ο΅ ) [ β Ο f ( g Ο ( Ο΅ ) ) ] β 1 N β i = 1 N β Ο f ( g Ο ( Ο΅ i ) ) {\displaystyle \nabla _{\phi }L(\phi )=\mat... | Wikipedia - Reparameterization trick - Mathematics > Reparameterization estimator | 233 | 477 | null |
Section: Examples. For some common distributions, the reparameterization trick takes specific forms: Normal distribution: For z βΌ N ( ΞΌ , Ο 2 ) {\displaystyle z\sim {\mathcal {N}}(\mu ,\sigma ^{2})} , we can use: z = ΞΌ + Ο Ο΅ , Ο΅ βΌ N ( 0 , 1 ) {\displaystyle z=\mu +\sigma \epsilon ,\quad \epsilon \sim {\mathcal {N}}(0,1... | Wikipedia - Reparameterization trick - Examples | 298 | 972 | null |
Section: Applications > Variational autoencoder. In Variational Autoencoders (VAEs), the VAE objective function, known as the Evidence Lower Bound (ELBO), is given by: ELBO ( Ο , ΞΈ ) = E z βΌ q Ο ( z | x ) [ log β‘ p ΞΈ ( x | z ) ] β D KL ( q Ο ( z | x ) | | p ( z ) ) {\displaystyle {\text{ELBO}}(\phi ,\theta )=\mathbb {E... | Wikipedia - Reparameterization trick - Applications > Variational autoencoder | 270 | 656 | null |
In Variational Autoencoders (VAEs), the VAE objective function, known as the Evidence Lower Bound (ELBO), is given by: ELBO ( Ο , ΞΈ ) = E z βΌ q Ο ( z | x ) [ log β‘ p ΞΈ ( x | z ) ] β D KL ( q Ο ( z | x ) | | p ( z ) ) {\displaystyle {\text{ELBO}}(\phi ,\theta )=\mathbb {E} _{z\sim q_{\phi }(z|x)}[\log p_{\theta }(x|z)]-... | Wikipedia - Reparameterization trick - Applications > Variational autoencoder | 437 | 1,018 | null |
The gradient of ELBO with respect to ΞΈ {\displaystyle \theta } is simply E z βΌ q Ο ( z | x ) [ β ΞΈ log β‘ p ΞΈ ( x | z ) ] β 1 L β l = 1 L β ΞΈ log β‘ p ΞΈ ( x | z l ) {\displaystyle \mathbb {E} _{z\sim q_{\phi }(z|x)}[\nabla _{\theta }\log p_{\theta }(x|z)]\approx {\frac {1}{L}}\sum _{l=1}^{L}\nabla _{\theta }\log p_{\thet... | Wikipedia - Reparameterization trick - Applications > Variational autoencoder | 357 | 859 | null |
Express the sampling operation z βΌ q Ο ( z | x ) {\displaystyle z\sim q_{\phi }(z|x)} as: z = ΞΌ Ο ( x ) + Ο Ο ( x ) β Ο΅ , Ο΅ βΌ N ( 0 , I ) {\displaystyle z=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon ,\quad \epsilon \sim {\mathcal {N}}(0,I)} where ΞΌ Ο ( x ) {\displaystyle \mu _{\phi }(x)} and Ο Ο ( x ) {\displaysty... | Wikipedia - Reparameterization trick - Applications > Variational autoencoder | 397 | 887 | null |
Then we have β Ο ELBO ( Ο , ΞΈ ) = E Ο΅ βΌ N ( 0 , I ) [ β Ο log β‘ p ΞΈ ( x | z ) + β Ο log β‘ q Ο ( z | x ) β β Ο log β‘ p ( z ) ] {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi ,\theta )=\mathbb {E} _{\epsilon \sim {\mathcal {N}}(0,I)}[\nabla _{\phi }\log p_{\theta }(x|z)+\nabla _{\phi }\log q_{\phi }(z|x)-\nabla _{\phi ... | Wikipedia - Reparameterization trick - Applications > Variational autoencoder | 524 | 1,070 | null |
This allows us to estimate the gradient using Monte Carlo sampling: β Ο ELBO ( Ο , ΞΈ ) β 1 L β l = 1 L [ β Ο log β‘ p ΞΈ ( x | z l ) + β Ο log β‘ q Ο ( z l | x ) β β Ο log β‘ p ( z l ) ] {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi ,\theta )\approx {\frac {1}{L}}\sum _{l=1}^{L}[\nabla _{\phi }\log p_{\theta }(x|z_{l})+... | Wikipedia - Reparameterization trick - Applications > Variational autoencoder | 341 | 803 | null |
Section: Applications > Variational inference. More generally, the trick allows using stochastic gradient descent for variational inference. Let the variational objective (ELBO) be of the form: ELBO ( Ο ) = E z βΌ q Ο ( z ) [ log β‘ p ( x , z ) β log β‘ q Ο ( z ) ] {\displaystyle {\text{ELBO}}(\phi )=\mathbb {E} _{z\sim q... | Wikipedia - Reparameterization trick - Applications > Variational inference | 331 | 818 | null |
Section: Applications > Dropout. The reparameterization trick has been applied to reduce the variance in dropout, a regularization technique in neural networks. The original dropout can be reparameterized with Bernoulli distributions: y = ( W β Ο΅ ) x , Ο΅ i j βΌ Bernoulli ( Ξ± i j ) {\displaystyle y=(W\odot \epsilon )x,\q... | Wikipedia - Reparameterization trick - Applications > Dropout | 164 | 528 | null |
The original dropout can be reparameterized with Bernoulli distributions: y = ( W β Ο΅ ) x , Ο΅ i j βΌ Bernoulli ( Ξ± i j ) {\displaystyle y=(W\odot \epsilon )x,\quad \epsilon _{ij}\sim {\text{Bernoulli}}(\alpha _{ij})} where W {\displaystyle W} is the weight matrix, x {\displaystyle x} is the input, and Ξ± i j {\displaysty... | Wikipedia - Reparameterization trick - Applications > Dropout | 372 | 938 | null |
More generally, other distributions can be used than the Bernoulli distribution, such as the gaussian noise: y i = ΞΌ i + Ο i β Ο΅ i , Ο΅ i βΌ N ( 0 , I ) {\displaystyle y_{i}=\mu _{i}+\sigma _{i}\odot \epsilon _{i},\quad \epsilon _{i}\sim {\mathcal {N}}(0,I)} where ΞΌ i = m i β€ x {\displaystyle \mu _{i}=\mathbf {m} _{i}^{\... | Wikipedia - Reparameterization trick - Applications > Dropout | 263 | 678 | null |
Article: Right to explanation. In the regulation of algorithms, particularly artificial intelligence and its subfield of machine learning, a right to explanation (or right to an explanation) is a right to be given an explanation for an output of the algorithm. Such rights primarily refer to individual rights to be give... | Wikipedia - Right to explanation - Summary | 258 | 1,415 | null |
Section: Examples > Credit scoring in the United States. Under the Equal Credit Opportunity Act (Regulation B of the Code of Federal Regulations), Title 12, Chapter X, Part 1002, Β§1002.9, creditors are required to notify applicants who are denied credit with specific reasons for the detail. As detailed in Β§1002.9(b)(2)... | Wikipedia - Right to explanation - Examples > Credit scoring in the United States | 254 | 1,233 | null |
Section: Examples > European Union. The European Union General Data Protection Regulation (enacted 2016, taking effect 2018) extends the automated decision-making rights in the 1995 Data Protection Directive to provide a legally disputed form of a right to an explanation, stated as such in Recital 71: "[the data subjec... | Wikipedia - Right to explanation - Examples > European Union | 341 | 1,852 | null |
In addition, there are significant restrictions on the types of automated decisions that are covered β which must be both "solely" based on automated processing, and have legal or similarly significant effects β which significantly limits the range of automated systems and decisions to which the right would apply. In p... | Wikipedia - Right to explanation - Examples > European Union | 348 | 1,863 | null |
Section: Examples > France. In France the 2016 Loi pour une RΓ©publique numΓ©rique (Digital Republic Act or loi numΓ©rique) amends the country's administrative code to introduce a new provision for the explanation of decisions made by public sector bodies about individuals. It notes that where there is "a decision taken o... | Wikipedia - Right to explanation - Examples > France | 266 | 1,355 | null |
Section: Criticism. Some argue that a "right to explanation" is at best unnecessary, at worst harmful, and threatens to stifle innovation. Specific criticisms include: favoring human decisions over machine decisions, being redundant with existing laws, and focusing on process over outcome. Authors of study βSlave to th... | Wikipedia - Right to explanation - Criticism | 285 | 1,476 | null |
To mitigate this issue, Edwards and Veale argue that an auditing system could be more effective, to allow auditors to look at the inputs and outputs of a decision process from an external shell, in other words, βexplaining black boxes without opening them.β Similarly, Oxford scholars Bryce Goodman and Seth Flaxman asse... | Wikipedia - Right to explanation - Criticism | 293 | 1,490 | null |
Section: Suggestions. Edwards and Veale see the right to explanation as providing some grounds for explanations about specific decisions. They discuss two types of algorithmic explanations, model centric explanations and subject-centric explanations (SCEs), which are broadly aligned with explanations about systems or d... | Wikipedia - Right to explanation - Suggestions | 340 | 1,891 | null |
Article: Robot learning. Robot learning is a research field at the intersection of machine learning and robotics. It studies techniques allowing a robot to acquire novel skills or adapt to its environment through learning algorithms. The embodiment of the robot, situated in a physical embedding, provides at the same ti... | Wikipedia - Robot learning - Summary | 251 | 1,382 | null |
Section: Sharing learned skills and knowledge. In Tellex's "Million Object Challenge," the goal is robots that learn how to spot and handle simple items and upload their data to the cloud to allow other robots to analyze and use the information. RoboBrain is a knowledge engine for robots which can be freely accessed by... | Wikipedia - Robot learning - Sharing learned skills and knowledge | 214 | 1,101 | null |
Article: Robotic process automation. Robotic process automation (RPA) is a form of business process automation that is based on software robots (bots) or artificial intelligence (AI) agents. RPA should not be confused with artificial intelligence as it is based on automation technology following a predefined workflow. ... | Wikipedia - Robotic process automation - Summary | 275 | 1,439 | null |
Section: Historic evolution. The typical benefits of robotic automation include reduced cost; increased speed, accuracy, and consistency; improved quality and scalability of production. Automation can also provide extra security, especially for sensitive data and financial services. As a form of automation, the concept... | Wikipedia - Robotic process automation - Historic evolution | 320 | 1,725 | null |
Section: Use. The hosting of RPA services also aligns with the metaphor of a software robot, with each robotic instance having its own virtual workstation, much like a human worker. The robot uses keyboard and mouse controls to take actions and execute automations. Normally, all of these actions take place in a virtual... | Wikipedia - Robotic process automation - Use | 212 | 1,139 | null |
Section: Impact on employment. According to Harvard Business Review, most operations groups adopting RPA have promised their employees that automation would not result in layoffs. Instead, workers have been redeployed to do more interesting work. One academic study highlighted that knowledge workers did not feel threat... | Wikipedia - Robotic process automation - Impact on employment | 270 | 1,364 | null |
Section: Impact on employment > Impact on society. Academic studies project that RPA, among other technological trends, is expected to drive a new wave of productivity and efficiency gains in the global labour market. Although not directly attributable to RPA alone, Oxford University conjectures that up to 35% of all j... | Wikipedia - Robotic process automation - Impact on employment > Impact on society | 340 | 1,758 | null |
He provides a case study of the Japanese insurance companies β Sompo Japan and Aioi β both of whom introduced bots to speed up the process of insurance pay-outs in past massive disaster incidents. Meanwhile, Professor Willcocks, author of the LSE paper cited above, speaks of increased job satisfaction and intellectual ... | Wikipedia - Robotic process automation - Impact on employment > Impact on society | 176 | 861 | null |
Section: Hyperautomation. Hyperautomation is the application of advanced technologies like RPA, artificial intelligence, machine learning (ML) and process mining to augment workers and automate processes in ways that are significantly more impactful than traditional automation capabilities. Hyperautomation is the combi... | Wikipedia - Robotic process automation - Hyperautomation | 166 | 837 | null |
Section: Outsourcing. Back office clerical processes outsourced by large organisations - particularly those sent offshore - tend to be simple and transactional in nature, requiring little (if any) analysis or subjective judgement. This would seem to make an ideal starting point for organizations beginning to adopt robo... | Wikipedia - Robotic process automation - Outsourcing | 326 | 1,697 | null |
A robotised automation can be hosted in a data centre in any jurisdiction and this has two major consequences for BPO providers. Firstly, for example, a sovereign government may not be willing or legally able to outsource the processing of tax affairs and security administration. On this basis, if robots are compared t... | Wikipedia - Robotic process automation - Outsourcing | 174 | 930 | null |
Article: ROCm. ROCm is an Advanced Micro Devices (AMD) software stack for graphics processing unit (GPU) programming. ROCm spans several domains, including general-purpose computing on graphics processing units (GPGPU), high performance computing (HPC), and heterogeneous computing. It offers several programming models:... | Wikipedia - ROCm - Summary | 173 | 738 | null |
Section: Software ecosystem > Third-party integration > Other Languages > Julia. Julia has the AMDGPU.jl package, which integrates with LLVM and selects components of the ROCm stack. Instead of compiling code through HIP, AMDGPU.jl uses Julia's compiler to generate LLVM IR directly, which is later consumed by LLVM to g... | Wikipedia - ROCm - Software ecosystem > Third-party integration > Other Languages > Julia | 188 | 729 | null |
Section: Components. There is one kernel-space component, ROCk, and the rest - there is roughly a hundred components in the stack - is made of user-space modules. The unofficial typographic policy is to use: uppercase ROC lowercase following for low-level libraries, i.e. ROCt, and the contrary for user-facing libraries... | Wikipedia - ROCm - Components | 153 | 648 | null |
Article: Rule induction. Rule induction is an area of machine learning in which formal rules are extracted from a set of observations. The rules extracted may represent a full scientific model of the data, or merely represent local patterns in the data. Data mining in general and rule induction in detail are trying to ... | Wikipedia - Rule induction - Summary | 191 | 993 | null |
Article: Sample complexity. The sample complexity of a machine learning algorithm represents the number of training-samples that it needs in order to successfully learn a target function. More precisely, the sample complexity is the number of training-samples that we need to supply to the algorithm, so that the functio... | Wikipedia - Sample complexity - Summary | 219 | 1,120 | null |
Section: Definition. Let X {\displaystyle X} be a space which we call the input space, and Y {\displaystyle Y} be a space which we call the output space, and let Z {\displaystyle Z} denote the product X Γ Y {\displaystyle X\times Y} . For example, in the setting of binary classification, X {\displaystyle X} is typicall... | Wikipedia - Sample complexity - Definition | 266 | 955 | null |
Fix a loss function L : Y Γ Y β R β₯ 0 {\displaystyle {\mathcal {L}}\colon Y\times Y\to \mathbb {R} _{\geq 0}} , for example, the square loss L ( y , y β² ) = ( y β y β² ) 2 {\displaystyle {\mathcal {L}}(y,y')=(y-y')^{2}} , where h ( x ) = y β² {\displaystyle h(x)=y'} . For a given distribution Ο {\displaystyle \rho } on X... | Wikipedia - Sample complexity - Definition | 458 | 1,017 | null |
For a given distribution Ο {\displaystyle \rho } on X Γ Y {\displaystyle X\times Y} , the expected risk of a hypothesis (a function) h β H {\displaystyle h\in {\mathcal {H}}} is E ( h ) := E Ο [ L ( h ( x ) , y ) ] = β« X Γ Y L ( h ( x ) , y ) d Ο ( x , y ) {\displaystyle {\mathcal {E}}(h):=\mathbb {E} _{\rho }[{\mathca... | Wikipedia - Sample complexity - Definition | 355 | 802 | null |
Define the optimal risk E H β = inf h β H E ( h ) . {\displaystyle {\mathcal {E}}_{\mathcal {H}}^{*}={\underset {h\in {\mathcal {H}}}{\inf }}{\mathcal {E}}(h).} Set h n = A ( S n ) {\displaystyle h_{n}={\mathcal {A}}(S_{n})} , for each sample size n {\displaystyle n} . h n {\displaystyle h_{n}} is a random variable and... | Wikipedia - Sample complexity - Definition | 345 | 893 | null |
In other words, for all Ο΅ , Ξ΄ > 0 {\displaystyle \epsilon ,\delta >0} , there exists a positive integer N {\displaystyle N} , such that, for all sample sizes n β₯ N {\displaystyle n\geq N} , we have Pr Ο n [ E ( h n ) β E H β β₯ Ξ΅ ] < Ξ΄ . {\displaystyle \Pr _{\rho ^{n}}[{\mathcal {E}}(h_{n})-{\mathcal {E}}_{\mathcal {H}}... | Wikipedia - Sample complexity - Definition | 336 | 929 | null |
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