REPORTS
larized nearly vertically. For completeness, Fig. metamaterials would be highly desirable but is
1B shows the off-resonant case for the smaller currently not available.
SRRs for vertical incident polarization.
References and Notes
Although these results are compatible with 1. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart,
the known selection rules of surface SHG from IEEE Trans. Microw. Theory Tech. 47, 2075 (1999).
usual nonlinear optics (23), these selection rules 2. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
do not explain the mechanism of SHG. Follow- 3. R. A. Shelby, D. R. Smith, S. Schultz, Science 292, 77 (2001).
4. T. J. Yen et al., Science 303, 1494 (2004).
ing our above argumentation on the magnetic 5. S. Linden et al., Science 306, 1351 (2004).
component of the Lorentz force, we numerically 6. C. Enkrich et al., Phys. Rev. Lett. 95, 203901 (2005).
calculate first the linear electric and magnet- 7. A. N. Grigorenko et al., Nature 438, 335 (2005).
ic field distributions (22); from these fields, 8. G. Dolling, M. Wegener, S. Linden, C. Hormann, Opt.
Express 14, 1842 (2006).
we compute the electron velocities and the
9. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis,
Lorentz-force field (fig. S1). In the spirit of a S. Linden, Science 312, 892 (2006).
metamaterial, the transverse component of the 10. J. B. Pendry, D. Schurig, D. R. Smith, Science 312, 1780;
Lorentz-force field can be spatially averaged published online 25 May 2006.
11. U. Leonhardt, Science 312, 1777 (2006); published
over the volume of the unit cell of size a by a
online 25 May 2006.
by t. This procedure delivers the driving force 12. M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis,
for the transverse SHG polarization. As usual, S. Linden, Opt. Lett. 31, 1259 (2006).
the SHG intensity is proportional to the square 13. W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, R. D. Averitt,
modulus of the nonlinear electron displacement. Phys. Rev. Lett. 96, 107401 (2006).
14. D. R. Smith, S. Schultz, P. Markos, C. M. Soukoulis, Phys.
Thus, the SHG strength is expected to be Rev. B 65, 195104 (2002).
proportional to the square modulus of the 15. S. O’Brien, D. McPeake, S. A. Ramakrishna, J. B. Pendry,
driving force, and the SHG polarization is Phys. Rev. B 69, 241101 (2004).
Fig. 3. Theory, presented as the experiment (see directed along the driving-force vector. Cor- 16. J. Zhou et al., Phys. Rev. Lett. 95, 223902 (2005).
Fig. 1). The SHG source is the magnetic compo- 17. A. K. Popov, V. M. Shalaev, available at http://arxiv.org/
responding results are summarized in Fig. 3 in abs/physics/0601055 (2006).
nent of the Lorentz force on metal electrons in the same arrangement as Fig. 1 to allow for a 18. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
the SRRs. direct comparison between experiment and 19. M. Wegener, Extreme Nonlinear Optics (Springer, Berlin,
theory. The agreement is generally good, both 2004).
20. H. M. Barlow, Nature 173, 41 (1954).
The setup for measuring the SHG is described for linear optics and for SHG. In particular, we
21. S.-Y. Chen, M. Maksimchuk, D. Umstadter, Nature 396,
in the supporting online material (22). We expect find a much larger SHG signal for excitation of 653 (1998).
that the SHG strongly depends on the resonance those two resonances (Fig. 3, A and C), which 22. Materials and Methods are available as supporting
that is excited. Obviously, the incident polariza- are related to a finite magnetic-dipole moment material on Science Online.
(perpendicular to the SRR plane) as compared 23. P. Guyot-Sionnest, W. Chen, Y. R. Shen, Phys. Rev. B 33,
tion and the detuning of the laser wavelength
8254 (1986).
from the resonance are of particular interest. One with the purely electric Mie resonance (Figs. 24. We thank the groups of S. W. Koch, J. V. Moloney, and
possibility for controlling the detuning is to 1D and 3D), despite the fact that its oscillator C. M. Soukoulis for discussions. The research of
change the laser wavelength for a given sample, strength in the linear spectrum is comparable. M.W. is supported by the Leibniz award 2000 of the
which is difficult because of the extremely broad The SHG polarization in the theory is strictly Deutsche Forschungsgemeinschaft (DFG), that of S.L. through
a Helmholtz-Hochschul-Nachwuchsgruppe (VH-NG-232).
tuning range required. Thus, we follow an vertical for all resonances. Quantitative devia-
Supporting Online Material
alternative route, lithographic tuning (in which tions between the SHG signal strengths of ex- www.sciencemag.org/cgi/content/full/313/5786/502/DC1
the incident laser wavelength of 1.5 mm, as well periment and theory, respectively, are probably Materials and Methods
as the detection system, remains fixed), and tune due to the simplified SRR shape assumed in Figs. S1 and S2
the resonance positions by changing the SRR our calculations and/or due to the simplicity of References
size. In this manner, we can also guarantee that our modeling. A systematic microscopic theory 26 April 2006; accepted 22 June 2006
the optical properties of the SRR constituent of the nonlinear optical properties of metallic 10.1126/science.1129198
materials are identical for all configurations. The
blue bars in Fig. 1 summarize the measured SHG
signals. For excitation of the LC resonance in Fig.
1A (horizontal incident polarization), we find Reducing the Dimensionality of
an SHG signal that is 500 times above the noise
level. As expected for SHG, this signal closely
scales with the square of the incident power
Data with Neural Networks
(Fig. 2A). The polarization of the SHG emission G. E. Hinton* and R. R. Salakhutdinov
is nearly vertical (Fig. 2B). The small angle with
respect to the vertical is due to deviations from High-dimensional data can be converted to low-dimensional codes by training a multilayer neural
perfect mirror symmetry of the SRRs (see network with a small central layer to reconstruct high-dimensional input vectors. Gradient descent
electron micrographs in Fig. 1). Small detuning can be used for fine-tuning the weights in such ‘‘autoencoder’’ networks, but this works well only if
of the LC resonance toward smaller wavelength the initial weights are close to a good solution. We describe an effective way of initializing the
(i.e., to 1.3-mm wavelength) reduces the SHG weights that allows deep autoencoder networks to learn low-dimensional codes that work much
signal strength from 100% to 20%. For ex- better than principal components analysis as a tool to reduce the dimensionality of data.
citation of the Mie resonance with vertical
incident polarization in Fig. 1D, we find a small imensionality reduction facilitates the finds the directions of greatest variance in the
signal just above the noise level. For excitation
of the Mie resonance with horizontal incident
polarization in Fig. 1C, a small but significant
D classification, visualization, communi-
cation, and storage of high-dimensional
data. A simple and widely used method is
data set and represents each data point by its
coordinates along each of these directions. We
describe a nonlinear generalization of PCA that
SHG emission is found, which is again po- principal components analysis (PCA), which uses an adaptive, multilayer Bencoder[ network
504 28 JULY 2006 VOL 313 SCIENCE www.sciencemag.org
REPORTS
to transform the high-dimensional data into a Starting with random weights in the two called an Bautoencoder[ and is depicted in
low-dimensional code and a similar Bdecoder[ networks, they can be trained together by Fig. 1.
network to recover the data from the code. minimizing the discrepancy between the orig- It is difficult to optimize the weights in
inal data and its reconstruction. The required nonlinear autoencoders that have multiple
Department of Computer Science, University of Toronto, 6 gradients are easily obtained by using the chain hidden layers (2–4). With large initial weights,
King’s College Road, Toronto, Ontario M5S 3G4, Canada. rule to backpropagate error derivatives first autoencoders typically find poor local minima;
*To whom correspondence should be addressed; E-mail: through the decoder network and then through with small initial weights, the gradients in the
hinton@cs.toronto.edu the encoder network (1). The whole system is early layers are tiny, making it infeasible to
train autoencoders with many hidden layers. If
the initial weights are close to a good solution,
Decoder
gradient descent works well, but finding such
30
initial weights requires a very different type of
W4 algorithm that learns one layer of features at a
Top
500 time. We introduce this Bpretraining[ procedure
RBM
T T
W1 W 1 +ε 8 for binary data, generalize it to real-valued data,
2000 2000 and show that it works well for a variety of
500
T
W2
T
W 2 +ε 7 data sets.
W3 1000 1000 An ensemble of binary vectors (e.g., im-
1000 T
W3
T
W 3 +ε 6 ages) can be modeled using a two-layer net-
RBM
500 500 work called a Brestricted Boltzmann machine[
T
W4 T
W 4 +ε 5
(RBM) (5, 6) in which stochastic, binary pixels
30 Code layer 30
are connected to stochastic, binary feature
1000 detectors using symmetrically weighted con-
W4 W 4 +ε 4
W2 nections. The pixels correspond to Bvisible[
2000 500 500
RBM units of the RBM because their states are
W3 W 3 +ε 3
observed; the feature detectors correspond to
1000 1000
Bhidden[ units. A joint configuration (v, h) of
W2 W 2 +ε 2 the visible and hidden units has an energy (7)
2000 2000 2000 given by
W1 W1 W 1 +ε 1 X X
Eðv, hÞ 0 j bi vi j bj hj
iZpixels jZfeatures
X ð1Þ
j vi hj wij
i, j
RBM Encoder
Pretraining Unrolling Fine-tuning where vi and hj are the binary states of pixel i
Fig. 1. Pretraining consists of learning a stack of restricted Boltzmann machines (RBMs), each and feature j, bi and bj are their biases, and wij
having only one layer of feature detectors. The learned feature activations of one RBM are used is the weight between them. The network as-
as the ‘‘data’’ for training the next RBM in the stack. After the pretraining, the RBMs are signs a probability to every possible image via
‘‘unrolled’’ to create a deep autoencoder, which is then fine-tuned using backpropagation of this energy function, as explained in (8). The
error derivatives. probability of a training image can be raised by
Fig. 2. (A) Top to bottom:
Random samples of curves from
the test data set; reconstructions
produced by the six-dimensional
deep autoencoder; reconstruc-
tions by ‘‘logistic PCA’’ (8) using
six components; reconstructions
by logistic PCA and standard
PCA using 18 components. The
average squared error per im-
age for the last four rows is
1.44, 7.64, 2.45, 5.90. (B) Top
to bottom: A random test image
from each class; reconstructions
by the 30-dimensional autoen-
coder; reconstructions by 30-
dimensional logistic PCA and
standard PCA. The average
squared errors for the last three
rows are 3.00, 8.01, and 13.87.
(C) Top to bottom: Random
samples from the test data set;
reconstructions by the 30-
dimensional autoencoder; reconstructions by 30-dimensional PCA. The average squared errors are 126 and 135.
www.sciencemag.org SCIENCE VOL 313 28 JULY 2006 505
REPORTS
adjusting the weights and biases to lower the the hidden units are then updated once more so same learning rule is used for the biases. The
energy of that image and to raise the energy of that they represent features of the confabula- learning works well even though it is not
similar, Bconfabulated[ images that the network tion. The change in a weight is given by exactly following the gradient of the log
would prefer to the real data. Given a training probability of the training data (6).
image, the binary state hj of each feature de- A single layer of binary features is not the
Dwij 0 e bvi hj Àdata j bvi hj Àrecon ð2Þ
tector
P j is set to 1 with probability s(bj þ best way to model the structure in a set of im-
iviwij), where s(x) is the logistic function ages. After learning one layer of feature de-
1/E1 þ exp (–x)^, bj is the bias of j, vi is the where e is a learning rate, bvi hjÀdata is the tectors, we can treat their activities—when they
state of pixel i, and wij is the weight between i fraction of times that the pixel i and feature are being driven by the data—as data for
and j. Once binary states have been chosen for detector j are on together when the feature learning a second layer of features. The first
the hidden units, a Bconfabulation[ is produced detectors are being driven by data, and layer of feature detectors then become the
P setting each vi to 1 with probability s(bi þ
by bvi hjÀrecon is the corresponding fraction for visible units for learning the next RBM. This
jhjwij), where bi is the bias of i. The states of confabulations. A simplified version of the layer-by-layer learning can be repeated as many
Fig. 3. (A) The two-
dimensional codes for 500
digits of each class produced
by taking the first two prin-
cipal components of all
60,000 training images.
(B) The two-dimensional
codes found by a 784-
1000-500-250-2 autoen-
coder. For an alternative
visualization, see (8).
Fig. 4. (A) The fraction of
retrieved documents in the
same class as the query when
a query document from the
test set is used to retrieve other
test set documents, averaged
over all 402,207 possible que-
ries. (B) The codes produced
by two-dimensional LSA. (C)
The codes produced by a 2000-
500-250-125-2 autoencoder.
506 28 JULY 2006 VOL 313 SCIENCE www.sciencemag.org
REPORTS
times as desired. It can be shown that adding an pi is the intensity of pixel i and ĝpi is the tion task, the best reported error rates are 1.6% for
extra layer always improves a lower bound on intensity of its reconstruction. randomly initialized backpropagation and 1.4%
the log probability that the model assigns to the The autoencoder consisted of an encoder for support vector machines. After layer-by-layer
training data, provided the number of feature with layers of size (28 28)-400-200-100- pretraining in a 784-500-500-2000-10 network,
detectors per layer does not decrease and their 50-25-6 and a symmetric decoder. The six backpropagation using steepest descent and a
weights are initialized correctly (9). This bound units in the code layer were linear and all the small learning rate achieves 1.2% (8). Pretraining
does not apply when the higher layers have other units were logistic. The network was helps generalization because it ensures that most
fewer feature detectors, but the layer-by-layer trained on 20,000 images and tested on 10,000 of the information in the weights comes from
learning algorithm is nonetheless a very effec- new images. The autoencoder discovered how modeling the images. The very limited informa-
tive way to pretrain the weights of a deep auto- to convert each 784-pixel image into six real tion in the labels is used only to slightly adjust
encoder. Each layer of features captures strong, numbers that allow almost perfect reconstruction the weights found by pretraining.
high-order correlations between the activities of (Fig. 2A). PCA gave much worse reconstruc- It has been obvious since the 1980s that
units in the layer below. For a wide variety of tions. Without pretraining, the very deep auto- backpropagation through deep autoencoders
data sets, this is an efficient way to pro- encoder always reconstructs the average of the would be very effective for nonlinear dimen-
gressively reveal low-dimensional, nonlinear training data, even after prolonged fine-tuning sionality reduction, provided that computers
structure. (8). Shallower autoencoders with a single were fast enough, data sets were big enough,
After pretraining multiple layers of feature hidden layer between the data and the code and the initial weights were close enough to a
detectors, the model is Bunfolded[ (Fig. 1) to can learn without pretraining, but pretraining good solution. All three conditions are now
produce encoder and decoder networks that greatly reduces their total training time (8). satisfied. Unlike nonparametric methods (15, 16),
initially use the same weights. The global fine- When the number of parameters is the same, autoencoders give mappings in both directions
tuning stage then replaces stochastic activities deep autoencoders can produce lower recon- between the data and code spaces, and they can
by deterministic, real-valued probabilities and struction errors on test data than shallow ones, be applied to very large data sets because both
uses backpropagation through the whole auto- but this advantage disappears as the number of the pretraining and the fine-tuning scale linearly
encoder to fine-tune the weights for optimal parameters increases (8). in time and space with the number of training
reconstruction. Next, we used a 784-1000-500-250-30 auto- cases.
For continuous data, the hidden units of the encoder to extract codes for all the hand-
first-level RBM remain binary, but the visible written digits in the MNIST training set (11).
References and Notes
units are replaced by linear units with Gaussian The Matlab code that we used for the pre- 1. D. C. Plaut, G. E. Hinton, Comput. Speech Lang. 2, 35
noise (10). If this noise has unit variance, the training and fine-tuning is available in (8). Again, (1987).
stochastic update rule for the hidden units all units were logistic except for the 30 linear 2. D. DeMers, G. Cottrell, Advances in Neural Information
remains the same and the update rule for visible units in the code layer. After fine-tuning on all Processing Systems 5 (Morgan Kaufmann, San Mateo, CA,
1993), pp. 580–587.
unit i is to sample from P a Gaussian with unit 60,000 training images, the autoencoder was 3. R. Hecht-Nielsen, Science 269, 1860 (1995).
variance and mean bi þ j hjwij. tested on 10,000 new images and produced 4. N. Kambhatla, T. Leen, Neural Comput. 9, 1493
In all our experiments, the visible units of much better reconstructions than did PCA (1997).
every RBM had real-valued activities, which (Fig. 2B). A two-dimensional autoencoder 5. P. Smolensky, Parallel Distributed Processing: Volume 1:
Foundations, D. E. Rumelhart, J. L. McClelland, Eds. (MIT
were in the range E0, 1^ for logistic units. While produced a better visualization of the data Press, Cambridge, 1986), pp. 194–281.
training higher level RBMs, the visible units than did the first two principal components 6. G. E. Hinton, Neural Comput. 14, 1711 (2002).
were set to the activation probabilities of the (Fig. 3). 7. J. J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 79, 2554
hidden units in the previous RBM, but the We also used a 625-2000-1000-500-30 auto- (1982).
8. See supporting material on Science Online.
hidden units of every RBM except the top one encoder with linear input units to discover 30- 9. G. E. Hinton, S. Osindero, Y. W. Teh, Neural Comput. 18,
had stochastic binary values. The hidden units dimensional codes for grayscale image patches 1527 (2006).
of the top RBM had stochastic real-valued that were derived from the Olivetti face data set 10. M. Welling, M. Rosen-Zvi, G. Hinton, Advances in Neural
states drawn from a unit variance Gaussian (12). The autoencoder clearly outperformed Information Processing Systems 17 (MIT Press, Cambridge,
MA, 2005), pp. 1481–1488.
whose mean was determined by the input from PCA (Fig. 2C).
11. The MNIST data set is available at http://yann.lecun.com/
that RBM_s logistic visible units. This allowed When trained on documents, autoencoders exdb/mnist/index.html.
the low-dimensional codes to make good use of produce codes that allow fast retrieval. We rep- 12. The Olivetti face data set is available at www.
continuous variables and facilitated compari- resented each of 804,414 newswire stories (13) cs.toronto.edu/ roweis/data.html.
sons with PCA. Details of the pretraining and as a vector of document-specific probabilities 13. The Reuter Corpus Volume 2 is available at http://
trec.nist.gov/data/reuters/reuters.html.
fine-tuning can be found in (8). of the 2000 commonest word stems, and we 14. S. C. Deerwester, S. T. Dumais, T. K. Landauer, G. W.
To demonstrate that our pretraining algo- trained a 2000-500-250-125-10 autoencoder on Furnas, R. A. Harshman, J. Am. Soc. Inf. Sci. 41, 391
rithm allows us to fine-tune deep networks half of the stories with the use P of the multiclass (1990).
efficiently, we trained a very deep autoen- cross-entropy error function E– i pi log ĝpi^ for 15. S. T. Roweis, L. K. Saul, Science 290, 2323 (2000).
16. J. A. Tenenbaum, V. J. de Silva, J. C. Langford, Science
coder on a synthetic data set containing the fine-tuning. The 10 code units were linear 290, 2319 (2000).
images of Bcurves[ that were generated from and the remaining hidden units were logistic. 17. We thank D. Rumelhart, M. Welling, S. Osindero, and
three randomly chosen points in two di- When the cosine of the angle between two codes S. Roweis for helpful discussions, and the Natural
mensions (8). For this data set, the true in- was used to measure similarity, the autoencoder Sciences and Engineering Research Council of Canada for
funding. G.E.H. is a fellow of the Canadian Institute for
trinsic dimensionality is known, and the clearly outperformed latent semantic analysis
Advanced Research.
relationship between the pixel intensities and (LSA) (14), a well-known document retrieval
the six numbers used to generate them is method based on PCA (Fig. 4). Autoencoders Supporting Online Material
highly nonlinear. The pixel intensities lie (8) also outperform local linear embedding, a www.sciencemag.org/cgi/content/full/313/5786/504/DC1
between 0 and 1 and are very non-Gaussian, recent nonlinear dimensionality reduction algo- Materials and Methods
Figs. S1 to S5
so we used logistic output units in the auto- rithm (15). Matlab Code
encoder, and the fine-tuning stage of the Layer-by-layer pretraining can also be used
learning
P minimized
P the cross-entropy error for classification and regression. On a widely used 20 March 2006; accepted 1 June 2006
E– i pi log p̂i – i(1 – pi ) log(1 – ĝpi)^, where version of the MNIST handwritten digit recogni- 10.1126/science.1127647
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