10707 Deep Learning Russ Salakhutdinov

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10707 Deep Learning Russ Salakhutdinov Machine Learning Department [email protected] http://www.cs.cmu.edu/~rsalakhu/10707/ Autoencoders Neural Networks Online Course • Disclaimer: Much of the material and slides for this lecture were borrowed from Hugo Larochelle’s class on Neural Networks: • Hugo’s class covers many other topics: convolutional networks, neural language model, Boltzmann machines, autoencoders, sparse coding, etc. • We will use his material for some of the other lectures. 2 Autoencoders Hugo Larochelle Departement´ d’informatique Universite´ de Sherbrooke [email protected] October 16, 2012 Abstract Math for my slides “Autoencoders”. Autoencoders • Hugo Larochelle AutoencodersDepartement´ h(x)= d’informatique g(a(x)) Universite´ de Sherbrooke = sigm(b + Wx) • Feed-forward neural [email protected] trained to reproduce its input at the output layer • DecoderOctober 16, 2012 x = o(a(x)) = sigm(c + W h(x)) Abstract ⇤ b b f(Mathx) forx myl(f slides(x)) “Autoencoders”. = (x x )2 l(f(x)) = For binary(x log(units x )+(1 x ) log(1 x )) • ⌘ k k − k − k k k − k − k P P • b b Encoder b b h(x)=g(a(x)) = sigm(b + Wx) • 3 x = o(a(x)) = sigm(c + W⇤h(x)) b b f(x) x l(f(x)) = (x x )2 l(f(x)) = (x log(x )+(1 x ) log(1 x )) • ⌘ k k − k − k k k − k − k P P b b b b 1 1 Autoencoders Feature Representation Feed-back, Feed-forward, generative, bottom-up top-down Decoder Encoder path Input Image • Details of what goes insider the encoder and decoder matter! • Need constraints to avoid learning an identity. 4 Autoencoders Binary Features z Decoder Encoder filters D filters W. WTz z=σ(Wx) Linear Sigmoid function function path Input Image x 5 Another Autoencoder Model Binary Features z Encoder filters W. σ(WTz) z=σ(Wx) Decoder Sigmoid filters WT function path Binary Input x • Need additional constraints to avoid learning an identity. • Relates to Restricted Boltzmann Machines. • Encoder and Decoder filters can be different. 6 Autoencoders Autoencoders Hugo Larochelle Hugo Larochelle Autoencoders Departement´ d’informatique Departement´ d’informatique Universite´ de Sherbrooke Universite´ de Sherbrooke Hugo Larochelle [email protected] [email protected]´ d’informatique October 16, 2012 Universite´ de Sherbrooke October 16, 2012 [email protected] Abstract October 16, 2012 Abstract Math for my slides “Autoencoders”. Math for my slides “Autoencoders”. • •Abstract h(x)=g(b + Wx) Math for my slides “Autoencoders”. h(x)=g(b + Wx) = sigm(b + Wx) = sigm(b + Wx) • • h(x)=g(a(x)) x = o(cLoss+ W⇤h (Functionx)) • = sigm(b + Wx) = sigm(c + W⇤h(x)) x = o(c + W h(x)) • Loss function for binary inputs ⇤ b = sigm(c + W⇤h(x)) 2 f(x) x l(f(x)) = (xk xk) l(f(x)) = (xk log(xk)+(1 xk) log(1 xk)) • ⌘ k• − − k − − b 2 P Ø Cross-entropyP error function (reconstruction loss) f(x) x l(f(x)) = (x x ) b b b x = • bo(a(x⌘)) k k − k = sigm(c + W⇤h(x)) P b b • Loss function for real-valued inputs b b f(x) x l(f(x)) = 1 (x x )2 l(f(x)) = (x log(x )+(1 x ) log(1 x )) • ⌘ 2 k k − k − k k k − k − k Ø (t) (t) (t) (t) l(f(xsum)) of =squaredx P differencesx (reconstruction loss) P a(x )b b b b • r Ø we use a linear activation− function at the output b (t) (t) b a(x ) = b + Wx (t) ( (t) h(x ) = sigm(7 a(x )) (t) ( (t) a(x ) = c + W>h(x ) ( x(t) = sigm(a(x(t))) b ( (t) (t) b (t) b (t) l(f(x )) = x x ra(x ) ( − (t) (t) b cl(f(x )) = a(x(t))l(f(x )) r ( rb (t) (t) (t) l(f(x )) = W (t) l(f(x )) rh(x ) ( b ra(x ) (t) ⇣ (t) ⌘ (t) (t) a(x(t))l(f(x )) = h(x(bt))l(f(x )) [...,h(x )j(1 h(x )j),...] r 1 ( r − (t) ⇣ (t) ⌘ l(f(x )) = (t) l(f(x )) rb ( ra(x ) (t) (t) (t) (t) (t) > l(f(x )) = (t) l(f(x )) x > + h(x ) (t) l(f(x )) rW ( ra(x ) ra(x ) ⇣ ⌘ ⇣ ⌘ b 1 W⇤ = W> • 1 Autoencoders Hugo Larochelle Departement´ d’informatique Universite´ de Sherbrooke Autoencoders [email protected] AutoencodersOctober 16, 2012 Hugo Larochelle Autoencoders Departement´ d’informatique AutoencodersUniversite´ de Sherbrooke Hugo LarochelleHugo Larochelle [email protected] Hugo LarochelleDepartement´ D d’informatiqueepartement´ Abstract d’informatiqueHugo Larochelle Departement´ d’informatique Math for my slides “Autoencoders”.Universite´ de SherbrookeUniversite´ D deepartement´ Sherbrooke d’informatiqueOctober 16, 2012 Universite´[email protected] de Sherbrooke Universite´ de Sherbrooke Hugo Larochelle [email protected]@[email protected] Departement´ d’informatique • October 16, 2012 Abstract October 16, 2012 Math for my slides “Autoencoders”. Universite´ de Sherbrooke October 16, 2012 Octoberh(x)= 16,g(a( 2012x)) Abstract [email protected] • = sigm(bAbstract+ Wx) Math for my slides “Autoencoders”. Math for my slides “Autoencoders”. h(x)=g(a(x)) Abstract = sigm(Octoberb + Wx) 16, 2012 • Abstract Math for my slides “Autoencoders”.• • h(x)=g(a(x)) Math for my slides “Autoencoders”. x = o(a(xh))(x)=g(a(x)) • = sigm(b + Wx) = sigm(b + Wx) • = sigm(c + W⇤h(x))x = o(a(x)) Abstract h(x)=g(a(x)) b b = sigm(c + W⇤h(x)) •• • Math for my slides “Autoencoders”. f(x) x l(f(x)) = (x x )2 l(f(x)) = (x log(x )+(1b x ) log(1b x )) =k sigm(xk =b +kof(Wx(ax(x) ))) x l(f(x)) = k x(xk = xo()a2k(x))l(f(x)) = k (x log(kx )+(1 x ) log(1 x )) • ⌘ − • ⌘ h(x)=− kg(ka−(x))k − − k k− k − k − k = sigm(c + W⇤h(x)) (t) (t) =(t) sigm(c + W⇤h(x)) P (t) l(f(x )) =PxP x P b •ba(x ) b b • b 2 b • r b = sigm(−b bb+ Wx) bb b f(x) x l(f(x)) = (xk xk) l(f(x)) = (xk log(xk)+(12 xk) log(1 xk)) • k f(x) x l(f(xkb)) = k(xk(t)xk) l(f(x)) = k (x(tk)log(xk()+(1t) xk) log(1 xk())t) • ⌘ − • ⌘ − a(x − ) −b= b +−−Wx a(x ) −= b +hWx−(x)=g(b + Wx) (t) (t) (t) (t) (t) (t) (t) ( (t) a(x(t))l(f(x )) = xP xx = o(a(ax(x))(t))l(f(xP)) = xP x ( P h(x ) = sigm(a(x )) • r b −b • r b −bb (t) b (t) b b = sigm(b + Wx) h(x ) = sigm(a(x )) (t) ( (t) = sigm(c +(t) W⇤h(x)) (t) ( (t) a(x (t)) = c + W>h(x ) • b ba(x ) = b + Wx(t) a(x ) = b +(Wxt) ( b ( b a(x ) = c + W( >h(x ) (t) (t) b b h(x(t)) = sigm(a(x(t))) ( h(x(t)) = sigm(a(xx(t))) = sigm(a(x )) 2 ( x =(t) o(a(x)) ( (t) ( f(x) x l(f(x)) = (x x ) l(f(x)) = (x (log(t) x )+(1 xx ) log(1(t) =x sigm((t))) a(x ))b (t) k k k k a(kx ) k= c + W>kh(x ) a(xk ) = c + W>h(x ) • ( (t) (t) (t) • ⌘ − Loss Function− ( − − (t) ( (t) (t) b =(t)a(x( sigm(t))l(f(xxc ))+ W==⇤ sigm(hx(xa())bx x)) b x = sigm(a(xr)) (( − P P ( b (t) (t) x = o(c + W⇤h(x)) • b b b (t) b b(t) b (t) clb(f(bx )) b = a(x(t))l(f(x )) (t) b • For both cases, the gradient a(x )l(f(x )) = x (t) x r (t) b (t) ( rb b (t()t) (t) (t) (t) (t) l(f(x )) = x x (t) l(f(xr)) = x b x 2 (a(x ) b − (t) (t) = sigm(c + W⇤h(x)) has a veryf(x simple) x form:rl(a(fx (x) ))a( =x ) k((x=k b(x−t+)kWx) r l(f(x)) = h((x(t))l(k(ft)(xx−k))log(x=k)+(1Wb a(x(t)x)lk(f)( log(1x )) xk)) ( (t)(rt) ((t) r • ⌘ (t()t) cl(f(−x )) (=tt)) cl(f(ax(x )) )−l(f=(x a())x(t))l(f(x )) − b − l(f(x ))b = (t) l(f(xb )) b c h(x ) r =a sigm((x ) a((x r)) rb ( rb (t) ⇣ (t) ⌘ (t) (t) (rt) (t)( rb(t) (t) a(x(t))l(f(x )) = (t) h(x(bt))2l(f(x )) [...,h(x )j(1 h(x )j),...] ( (t) (t) l(f(x )) = W (t) (t) l(f(x )) (t) l(f(x )) =(t(x)tP) (t) x h(xf(()t(xt))) xr l((Ptf) (xb)) =a(x )((xk r xk) − a(x ) (t) l(fa((xx h)))(x )=l=(fW(xc +))W(t)rhl(=f(x(x W)))b a(x( )l(f(xr )) k • r b h(x ) −b b a(x >)• ⌘ (t) b ⇣− (t) ⌘ b r r (( r ( (tr) bl(f(⇣x )) =(t) a⌘(x(t))l(f((tx) )) (t) (t) (t)(t) l(f(x )) = (bt) l(f(x )) [...,h(x ) (1 h(x ) ),...] (xt) = sigm(⇣(t) a(xa(x(t))) ) ⌘⇣ r(t) (th)(x ) (t⌘)( r (t) j (t) j (t) l(f(x )) = (bt) l(rf(x )) [...,h((x ) (1r h(xP) ),...] − b a(x ) a(x(t())l(f(xh(x))) = (ht()x(bt))l(f(jx ())t) [j...,h(t)(x )j(1 (ht)(x ()tj) ),...] (t) (t) > r ( r a(x (t) ) =⇣ −b + Wx(t) ⌘ (t) > (t) b rb ( bl(fr(x ))b =Wl(fa((xx(t))))l(f(x =))b a(x )l(f(−x )) x + h(x ) a(x )l(f(x )) (t) ⇣ (t)r ⌘ (r( r ( r r • Parameter gradients areb obtainedl(f(x )) by =backpropagatinga((xt()t))l(f(x )) the⇣ gradient(t) (t) ⌘ (t) l(f(x )) = (t)(t) l(f(x )) (t) ⇣ (t) (t) ⌘ (t) > ⇣ ⌘ (t) r (t) b (t()b r b hl((fx(ax(x )))) = = sigm((t) l(f(xa())x x ))> + h(x ) (t) l(f(x )) b a ( x ( t ) ) l ( f ( x )) like =in a xregularx networkr ( W r a(x ) a(x ) (t) r(t) (t)> (((t) r (t) > r r ( Wl(f(x −)) = a(x(t))l(f(x )) x (t) + h(x ⇣) a(x(t))l(f(x ⌘)) (t) ⇣ ⌘ (t) r ( (t)r(t) W⇤ =aW(x> ) = (tr)c + W(t)> h(x (t) b (t) > (t) l(f(x )) = (t) l(f(x )) x >+ h(x ) (t) l(f(x )) b cl(f(x )) = a(x )l(Wf(x ⇣)) • ⌘ a(x ) ⇣ ⌘ a(x ) r ( r r ( r(t) ( b r Ø important: when using btied weights ( W ⇤ = W > ), W l ( f ( x ))( t ) (t) (t) • (t) • r ⇣ x = sigm(⌘ a(x )) ⇣ ⌘ (t) l(f(x )) = Wb (t) l(f(x )) b h(x isW) the⇤ = sumW> of two gradients a(x ) ( r • ( r b Ø this is because(t)(t) is present⇣ in the encoder(t) and⌘ in the decoder(t) (t) (t) lW(fl((xf(x )))) W= (bt) l(f(x )) [...,h(x )j(1 h(x )j),...1] ra(x• r) ( rh(x ) (t) (t) b (t−) b 1 (t) l(f(x )) = x x (t) a⇣(x ) (t) ⌘ bl(f(x )) =r a(x(t))l(f(x )) ( − 8 r ( r (t) (t) cl(f(x )) =1 a(x(t))l(f(x )) (t) b (t) (t)> (t) 1 (t) > Wl(f(x )) = ra(x(t))l(f(x )) x( +rbh(x ) a(x(t))l(f(x )) r ( r (t) r (t) h⇣(x(t))l(f(x )) ⌘ = Wb a⇣(x(t))l(f(x )) ⌘ r ( r b (t) ⇣ (t) ⌘ (t) (t) (t) l(f(x )) = (bt) l(f(x )) [...,h(x ) (1 h(x ) ),...] ra(x ) ( rh(x ) j − j (t) ⇣ (t) ⌘ bl(f(x )) = a(x(t))l(f(x )) r 1 ( r (t) (t) (t) (t) (t) > l(f(x )) = (t) l(f(x )) x > + h(x ) (t) l(f(x )) rW ( ra(x ) ra(x ) ⇣ ⌘ ⇣ ⌘ b 1 1 Autoencoder (t) (t) • Adapting an autoencoder to a newWl (typef(x of ))inputW Wl(f(x )) W l(f(x(t))) W• r • r (t) • rW Wl(f(x )) W Ø choose a joint distribution p ( x µ ) overµ the inputs,p• (wherexrµ) µ p(x µ) µ • | (t)
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    LP-WaveNet: Linear Prediction-based WaveNet Speech Synthesis Min-Jae Hwang Frank Soong Eunwoo Song Search Solution Microsoft Naver Corporation Seongnam, South Korea Beijing, China Seongnam, South Korea [email protected] [email protected] [email protected] Xi Wang Hyeonjoo Kang Hong-Goo Kang Microsoft Yonsei University Yonsei University Beijing, China Seoul, South Korea Seoul, South Korea [email protected] [email protected] [email protected] Abstract—We propose a linear prediction (LP)-based wave- than the speech signal, the training and generation processes form generation method via WaveNet vocoding framework. A become more efficient. WaveNet-based neural vocoder has significantly improved the However, the synthesized speech is likely to be unnatural quality of parametric text-to-speech (TTS) systems. However, it is challenging to effectively train the neural vocoder when the target when the prediction errors in estimating the excitation are database contains massive amount of acoustical information propagated through the LP synthesis process. As the effect such as prosody, style or expressiveness. As a solution, the of LP synthesis is not considered in the training process, the approaches that only generate the vocal source component by synthesis output is vulnerable to the variation of LP synthesis a neural vocoder have been proposed. However, they tend to filter. generate synthetic noise because the vocal source component is independently handled without considering the entire speech To alleviate this problem, we propose an LP-WaveNet, production process; where it is inevitable to come up with a which enables to jointly train the complicated interactions mismatch between vocal source and vocal tract filter.
  • Lecture 11 Recurrent Neural Networks I CMSC 35246: Deep Learning

    Lecture 11 Recurrent Neural Networks I CMSC 35246: Deep Learning

    Lecture 11 Recurrent Neural Networks I CMSC 35246: Deep Learning Shubhendu Trivedi & Risi Kondor University of Chicago May 01, 2017 Lecture 11 Recurrent Neural Networks I CMSC 35246 Introduction Sequence Learning with Neural Networks Lecture 11 Recurrent Neural Networks I CMSC 35246 Some Sequence Tasks Figure credit: Andrej Karpathy Lecture 11 Recurrent Neural Networks I CMSC 35246 MLPs only accept an input of fixed dimensionality and map it to an output of fixed dimensionality Great e.g.: Inputs - Images, Output - Categories Bad e.g.: Inputs - Text in one language, Output - Text in another language MLPs treat every example independently. How is this problematic? Need to re-learn the rules of language from scratch each time Another example: Classify events after a fixed number of frames in a movie Need to resuse knowledge about the previous events to help in classifying the current. Problems with MLPs for Sequence Tasks The "API" is too limited. Lecture 11 Recurrent Neural Networks I CMSC 35246 Great e.g.: Inputs - Images, Output - Categories Bad e.g.: Inputs - Text in one language, Output - Text in another language MLPs treat every example independently. How is this problematic? Need to re-learn the rules of language from scratch each time Another example: Classify events after a fixed number of frames in a movie Need to resuse knowledge about the previous events to help in classifying the current. Problems with MLPs for Sequence Tasks The "API" is too limited. MLPs only accept an input of fixed dimensionality and map it to an output of fixed dimensionality Lecture 11 Recurrent Neural Networks I CMSC 35246 Bad e.g.: Inputs - Text in one language, Output - Text in another language MLPs treat every example independently.
  • A Guide to Recurrent Neural Networks and Backpropagation

    A Guide to Recurrent Neural Networks and Backpropagation

    A guide to recurrent neural networks and backpropagation Mikael Bod´en¤ [email protected] School of Information Science, Computer and Electrical Engineering Halmstad University. November 13, 2001 Abstract This paper provides guidance to some of the concepts surrounding recurrent neural networks. Contrary to feedforward networks, recurrent networks can be sensitive, and be adapted to past inputs. Backpropagation learning is described for feedforward networks, adapted to suit our (probabilistic) modeling needs, and extended to cover recurrent net- works. The aim of this brief paper is to set the scene for applying and understanding recurrent neural networks. 1 Introduction It is well known that conventional feedforward neural networks can be used to approximate any spatially finite function given a (potentially very large) set of hidden nodes. That is, for functions which have a fixed input space there is always a way of encoding these functions as neural networks. For a two-layered network, the mapping consists of two steps, y(t) = G(F (x(t))): (1) We can use automatic learning techniques such as backpropagation to find the weights of the network (G and F ) if sufficient samples from the function is available. Recurrent neural networks are fundamentally different from feedforward architectures in the sense that they not only operate on an input space but also on an internal state space – a trace of what already has been processed by the network. This is equivalent to an Iterated Function System (IFS; see (Barnsley, 1993) for a general introduction to IFSs; (Kolen, 1994) for a neural network perspective) or a Dynamical System (DS; see e.g.
  • Feedforward Neural Networks and Word Embeddings

    Feedforward Neural Networks and Word Embeddings

    Feedforward Neural Networks and Word Embeddings Fabienne Braune1 1LMU Munich May 14th, 2017 Fabienne Braune (CIS) Feedforward Neural Networks and Word Embeddings May 14th, 2017 · 1 Outline 1 Linear models 2 Limitations of linear models 3 Neural networks 4 A neural language model 5 Word embeddings Fabienne Braune (CIS) Feedforward Neural Networks and Word Embeddings May 14th, 2017 · 2 Linear Models Fabienne Braune (CIS) Feedforward Neural Networks and Word Embeddings May 14th, 2017 · 3 Binary Classification with Linear Models Example: the seminar at < time > 4 pm will Classification task: Do we have an < time > tag in the current position? Word Lemma LexCat Case SemCat Tag the the Art low seminar seminar Noun low at at Prep low stime 4 4 Digit low pm pm Other low timeid will will Verb low Fabienne Braune (CIS) Feedforward Neural Networks and Word Embeddings May 14th, 2017 · 4 Feature Vector Encode context into feature vector: 1 bias term 1 2 -3 lemma the 1 3 -3 lemma giraffe 0 ... ... ... 102 -2 lemma seminar 1 103 -2 lemma giraffe 0 ... ... ... 202 -1 lemma at 1 203 -1 lemma giraffe 0 ... ... ... 302 +1 lemma 4 1 303 +1 lemma giraffe 0 ... ... ... Fabienne Braune (CIS) Feedforward Neural Networks and Word Embeddings May 14th, 2017 · 5 Dot product with (initial) weight vector 2 3 2 3 x0 = 1 w0 = 1:00 6 x = 1 7 6 w = 0:01 7 6 1 7 6 1 7 6 x = 0 7 6 w = 0:01 7 6 2 7 6 2 7 6 ··· 7 6 ··· 7 6 7 6 7 6x = 17 6x = 0:017 6 101 7 6 101 7 6 7 6 7 6x102 = 07 6x102 = 0:017 T 6 7 6 7 h(X )= X · Θ X = 6 ··· 7 Θ = 6 ··· 7 6 7 6 7 6x201 = 17 6x201 = 0:017 6 7