{"title": "Reverse TDNN: An Architecture For Trajectory Generation", "book": "Advances in Neural Information Processing Systems", "page_first": 579, "page_last": 588, "abstract": null, "full_text": "Reverse TDNN: An Architecture for Trajectory \n\nGeneration \n\nPatrice Simard \n\nAT &T Bell Laboratories \n101 Crawford Corner Rd \n\nHolmdel, NJ 07733 \n\nYann Le Cun \n\nAT&T Bell Laboratories \n101 Crawford Corner Rd \n\nHolmdel, NJ 07733 \n\nAbstract \n\nThe backpropagation algorithm can be used for both recognition and gen(cid:173)\neration of time trajectories. When used as a recognizer, it has been shown \nthat the performance of a network can be greatly improved by adding \nstructure to the architecture. The same is true in trajectory generation. \nIn particular a new architecture corresponding to a \"reversed\" TDNN is \nproposed. Results show dramatic improvement of performance in the gen(cid:173)\neration of hand-written characters. A combination of TDNN and reversed \nTDNN for compact encoding is also suggested. \n\n1 \n\nINTRODUCTION \n\nTrajectory generation finds interesting applications in the field of robotics, automa(cid:173)\ntion, filtering, or time series prediction. Neural networks, with their ability to learn \nfrom examples, have been proposed very early on for solving non-linear control prob(cid:173)\nlems adaptively. Several neural net architectures have been proposed for trajectory \ngeneration, most notably recurrent networks, either with discrete time and exter(cid:173)\nnalloops (Jordan, 1986), or with continuous time (Pearlmutter, 1988). Aside from \nbeing recurrent, these networks are not specifically tailored for trajectory genera(cid:173)\ntion. It has been shown that specific architectures, such as the Time Delay Neural \nNetworks (Lang and Hinton, 1988), or convolutional networks in general, are better \nthan fully connected networks at recognizing time sequences such as speech (Waibel \net al., 1989), or pen trajectories (Guyon et al., 1991). We show that special archi(cid:173)\ntectures can also be devised for trajectory generation, with dramatic performance \nimprovement. \n\n579 \n\n\f580 \n\nSimard and Le Cun \n\nTwo main ideas are presented in this paper. The first one rests on the assumption \nthat most trajectory generation problems deal with continuous trajectories. Fol(cid:173)\nlowing (Pearlmutter, 1988), we present the \"differential units\", in which the total \ninput to the neuron controls the em rate of change (time derivative) of that unit \nstate, instead of directly controlling its state. As will be shown the \"differential \nunits\" can be implemented in terms of regular units. \nThe second idea comes from the fact that trajectories are usually come from a \nplan, resulting in the execution of a \"motor program\". Executing a complete motor \nprogram will typically involve executing a hierarchy of sub-programs, modified by \nthe information coming from sensors. For example drawing characters on a piece \nof paper involves deciding which character to draw (and what size), then drawing \neach stroke of the character. Each stroke involves particular sub-programs which \nare likely to be common to several characters (straight lines of various orientations, \ncurved lines, loops ... ). Each stroke is decomposed in precise motor patterns. In \nshort, a plan can be described in a hierarchical fashion, starting from the most \nabstract level (which object to draw), which changes every half second or so, to \nthe lower level (the precise muscle activation patterns) which changes every 5 or \n10 milliseconds. It seems that this scheme can be particularly well embodied by \nan \"Oversampled Reverse TDNN\". a multilayer architecture in which the states \nof the units in the higher layers are updated at a faster rate than the states of \nunits in lower layers. The ORTDNN resembles a Subsampled TDNN (Bottou et al., \n1990)(Guyon et al., 1991), or a subsampled weight-sharing network (Le Cun et al., \n1990a), in which all the connections have been reversed, and the input and output \nhave been interchanged. The advantage of using the ORTDNN, as opposed to a \ntable lookup, or a memory intensive scheme, is the ability to generalize the learned \ntrajectories to unseen inputs (plans). With this new architecture it is shown that \ntrajectory generation problems of large complexity can be solved with relatively \nsmall resources. \n\n2 THE DIFFERENTIAL UNITS \n\nIn a time continuous network, the forward propagation can be written as: \n\n8x(t) \n\nT{jt = -x(t) + g(wx(t\u00bb + I(t) \n\n(1) \n\nwhere x(t) is the activation vector for the units, T is a diagonal matrix such that \nni is the time constant for unit i, It is the input vector at time t, w is a weight \nmatrix such that Wij is the connection from unit j to unit i, and 9 is a differentiable \n(multi-valued) function. \n\nA reasonable discretization of this equation is: \n\nwhere ~t is the time step used in the discretization, the superscript t means at time \nt~t (i.e. xt = x(t~t\u00bb. Xo is the starting point and is a constant. t ranges from 0 \nto M, with 10 = o. \n\n(2) \n\n\fReverse TDNN: An Architecture for Trajectory Generarion \n\n581 \n\nThe cost function to be minimized is: \n\nt=M \n\nE = ~ L: (stxt - dt)T (stxt - dt) \n\nt=1 \n\n(3) \n\nWhere Dt is the desired output, and st is a rectangular matrix which has a 0 if \n\nthe corresponding x: is unconstrained and 1 otherwise. Each pattern is composed \nof pairs (It, Dt) for t E [1..M]. To minimize equation 3 with the constraints given \nby equation 2 we express the Lagrage function (Le Cun, 1988): \nL = ~ L:(Stxt_Dt)(Stxt_Dt)T + L: (bt+l)T(_xt+l+xt+LltT-l(_xt+g(wxt)+It\u00bb) \n\nt=M-l \n\nt=M \n\nt=1 \n\nt=O \n\n(4) \nWhere bt+l are Lagrange multipliers (for t E [1..MD. The superscript T means that \nthe corresponding matrix is transposed. If we differentiate with respect to xt we \nget: \n(:~ ) T = 0 = (sti' _ d') _ ii' + ;;'+1 _ ~tT-1ii'+1 _ ~tT-1wT g'(wi')ii'+1 (5) \n\nFor t E [l..M - 1] and 8~'t, = 0 = (S'x M - DM) - bM for the boundary condition. \ng' a diagonal matrix containing the derivatives of 9 (g'(wx)w is the jacobian of g). \nFrom this an update rule for bt can be derived: \nbM \n\n(SMXM _ dM) \n(S'xt - dt) + (1 - LltT-l)bt+l + LltT-lwTyrg(wxt)bt+l \n\nfor t E [1..M - 1] \n\n(6) \nThis is the rule used to compute the gradient (backpropagation). If the Lagrangian \nis differentiated with respect to Wij, the standard updating rule for the weight is \nobtained: \n\n(7) \n\n(8) \n\noL \now .. = LltT- 1 L: b;+lxjg;(L: wil:xi) \nIf the Lagrangian is differentiated with respect to T, we get: \n\nt=M-l_ \n\nt=1 \n\n~ \n\nl: \n\nt=M-l \n\noL _ T- 1 ~ (-t+l \n- - -\noT \n\nL.J x \nt=O \n\n-t)b-t+l \n\n-x \n\nFrom the last two equations, we can derived a learning algorithm by gradient descent \n\n(9) \n\n(10) \n\nwhere 7]w and 7]T are respectively the learning rates for the weights and the time \nconstants (in practice better results are obtained by having different learning rates \n7]Wjj and 7]Tii per connections). The constant 7]T must be chosen with caution \n\n\f582 \n\nSimard and Le Cun \n\nFigure 1: A backpropagation implementation of equation 2 for a two units network \nbetween time t and t + 1. This figure rer.eats itself vertically for every time step \nfrom t = 0 to t = M. The quantities x /1, x~+l, d~ = -x~ + gl (wxt) + If and \nd~ = -x~ + g2(wxt) + n are computed with linear units. \n\nII \n\nsince if any time constants tii were to become less than one, the system would \ninstead of in tii is preferable for \nbe unstable. Performing gradient descent in Tl \nnumerical stability reasons. \nEquation 2 is implemented with a feed forward backpropagation network. It should \nfirst be noted that this equation can be written as a linear combination of xt (the \nactivation at the previous time), the input, and a non-linear function g of wx'. \nTherefore, this can be implemented with two linear units and one nonlinear unit \nwith activation function g. To keep the time constraint, the network is \"unfolded\" \nin time , with the weights shared from one time step to another. For instance a \nsimple two fully connected units network with no threshold can be implemented \nas in Fig. 1 (only the layer between time t and t + 1 is shown). The network \nrepeats itself vertically for each time step with the weights shared between time \nsteps. The main advantage of this implementation is that all equations 6, 7 and 8 \nare implemented implicitly by the back-propagation algorithm. \n\n3 CHARACTER GENERATION: LEARNING TO \n\nGENERATE A SINGLE LETTER \n\nIn this section we describe a simple experiment designed to 1) illustrate how tra(cid:173)\njectory generation can be implemented with a recurrent network, 2) to show the \nadvantages of using differential units instead of the traditional non linear units and \n3) to show how the fully connected architecture (with differential units) severly \nlimits the learning capacity of the network. The task is to draw the letter \"A\" with \n\n\fReverse TDNN: An Architecture for Trajectory Generation \n\n583 \n\nTarget drawing \n\nOutput trajectories \n\n1.25 \n\n.15 \n\n.25 \n\n-.25 \n\n-.15 \n\n-1.25 ~ ____ __ _ _ \n\n-1.25 -.15 -.25 \n\n.25 \nOulpAl \n\n.15 1.25 \n\nNetworK drawing \n\n1.25 \n\n.15 \n\n.25 \n\n- .25 \n\n-.15 \n\n0Jtpu12 \n\n-1.25\"__ ______ _ \n\n-1.25 -.75 \n\n- . 25 \n\n. 25 \nOulpAl \n\n. 15 1.25 \n\n1.25 \n\n.15 \n\n.25 \n\n- . 25 \n\nOu1pJtO \n\n-.15 \n\n-1.25 ___ ____ _ \n\no 15 30 45 60 15 '0105120135 \n\n1.25 \n\n.15 \n\n.25 \n\n-.25 \n\n-.15 \n\n-1.25'--______ _ \n\no 15 )0 45 60 15 '0 105120135 \n\n1.25 \n\n.15 \n\n.25 \n\n-.25 \n\n- . 15 \n\nTime \n\nFigure 2: Top left: Trajectory representing the letter \"A\". Bottom left: Trajectory \nproduced by the network after learning. The dots correspond to the target points of \nthe original trajectory. The curve is produced by drawing output unit 2 as a function \nof output unit 1, using output unit 0 for deciding when the pen is up or down. Right: \nTrajectories of the three output units (pen-up/pen-down, X coordinate of the pen \nand Y coordinate of the pen) as a function of time. The dots corresponds to the \ntarget points of the original trajectory. \n\na pen. The network has 3 output units, two for the X and Y position of the pen, \nand one to code whether the pen is up or down. The network has a total 21 units, \nno input unit, 18 hidden units and 3 output units. The network is fully connected. \n\nCharacter glyphs are obtained from a tablet which records points at successive \ninstants of time. The data therefore is a sequence of triplets indicating the time, \nand the X and Y positions. When the pen is up, or if there are no constraint for \nsome specific time steps (misreading of the tablet), the activation of the unit is left \nunconstrained. The letter to be learned is taken from a handwritten letter database \nand is displayed in figure 2 (top left). \n\nThe letter trajectory covers a maximum of 90 time stamps. The network is unfolded \n135 steps (10 unconstrained steps are left at the begining to allow the network to \nsettle and 35 additional steps are left at the end to monitor the network activity). \nThe learning rate 'f/w is set to 1.0 (the actual learning rate is per connection and is \nobtained by dividing the global learning rate by the fanin to the destination unit, \nand by dividing by the number of connections sharing the same weight). The time \nconstants are set to 10 to produce a smooth trajectory on the output. The learning \nrate 'f/T is equal to zero (no learning on the time constants). The initial values for \nthe weights are picked from a uniform distribution between -1 and +1. \n\n\f584 \n\nSimard and Le Cun \n\nThe trajectories fo units 0, 1 and 2 are shown in figure 2 (right). The top graphs \nrepresent the state of the pen as a function of time. The straight lines are the desired \npositions (1 means pen down, -1 means pen up). The middle and bottom graphs \nare the X and Y positions of the pen respectively. The network is unconstrained \nafter time step 100. Even though the time constants are large, the output units \nreach the right values before time step 10. The top trajectory (pen-up/pen-down), \nhowever, is difficult to learn with time constants as large as 10 because it is not \nsmooth. \n\nThe letter drawn by the network after learning is shown in figure 2 (left bottom). \nThe network successfully learned to draw the letter on the fully connected network. \nDifferent fixed time constants were tried. For small time constant (like 1.0), the \nnetwork was unable to learn the pattern for any learning rate TJw we tried. This \nis not surprising since the (vertical) weight sharing makes the trajectories very \nsensitive to any variation of the weights. This fact emphasizes the importance of \nusing differential units. Larger time constants allow larger learning rate for the \nweights. Of course, if those are too large, fast trajectories can not be learned. \n\nThe error can be further improved by letting the time constant adapt as well. \nHowever the gain in doing so is minimal. If the learning rate TJT is small, the gain \nover 'TJT = 0 is negligible. If TJT is too big, learning becomes quickly unstable. \nThis simulation was done with no input, and the target trajectories were for the \ndrawing of a single letter. In the next section, the problem is extended to that of \nlearning to draw multiple letters, depending on an input vector. \n\n4 LEARNING TO GENERATE MULTIPLE LETTERS: \n\nTHE REVERSE TDNN ARCHITECTURE \n\nIn a first attempt, the fully connected network of the previous section was used to \ntry to generate the eight first letters of the alphabet. Eight units were used for \nthe input, 3 for the output, and various numbers of hidden units were tried. Every \ntime, all the units, visible and hidden, were fully interconnected. Each input unit \nwas associated to one letter, and the input patterns consisted of one +1 at the \nunit corresponding to the letter, and -1/7 for all other input units. No success was \nachieved for all the set of parameters which were tried. The error curves reached \nplateaus, and the letter .glyphs were not recognizable. Even bringing the number of \nletter to two (one \"A\" and one \"B\") was unsuccessful. In all cases the network was \nacting like it was ignoring its input: the activation of the output units were almost \nidentical for all input patterns. This was attributed to the network architecture. \n\nA new kind of architecture was then used, which we call\" Oversampled Reverse \nTDNN\" because of its resemblance with a Subsampled TDNN with input and out(cid:173)\nput interchanged. Subsampled TDNN have been used in speech recognition (Bottou \net al., 1990), and on-line character recognition (Guyon et al., 1991). They can be \nseen one-dimensional versions of locally-connected, weight sharing networks (Le \nCun, 1989 )(Le Cun et al., 1990b). Time delay connections allow units to be con(cid:173)\nnected to unit at an earlier time. Weight sharing in time implements a convolution \nof the input layer. In the Subsampled TDNN, the rate at which the units states \nare updated decreases gradually with the layer index. The subsampling provides \n\n\fReverse TDNN: An Architecture for Trajectory Generation \n\n585 \n\nt=13 \n\nt=5 \n\nInput \n\nHidden1 \n\nHidden 2 \n\nOutput \n\nFigure 3: Architecture of a simple reverse TDNN. Time goes from bottom to top, \ndata flows from left to right. The left module is the input and has 2 units. The \nnext module (hidden!) has 3 units and is undersampled every 4 time steps. The \nfollowing module (hidden2) has 4 units and is undersampled every 2 time steps. The \nright module is the output, has 3 units and is not undersampled. All modules have \ntime delay connections from the preceding module. Thus the hidden! is connected \nto hidden2 over a window of 5 time steps, and hidden2 to the output over a window \nof 3 time steps. For each pattern presented on the 2 input units, a trajectory of 8 \ntime steps is produced by the network on each of the 3 units of the output. \n\n\f586 \n\nSimard and Le Cun \n\n\u2022 \n\n\u2022 \n\n'. \n\n~ \n\n\u2022 \n\n\u2022 \n\nI \n\n\u2022 \n\nI \n\n\u2022 \n\n\u2022 \n\n\u2022 \n\n\u2022 \n\n\u2022 \n\n-I \n\n.. \n\n-I \n\nI \n\n'. \n\n\u2022 \n\nI \n\n_I _. _. \n\n\u2022 \n\n_I \n\n\u2022\u2022 - . \n\n\u2022 \n\nI \n\n... _. _. \n\n\u2022 \n\n... \n\n\u00b71 _J \u2022 \u2022 I \n\n..... _. _. \n\n\u2022 \n\n... - . - \u2022 \u2022 \u2022 I \n\n... - . - \u2022 \u2022 \u2022 I \n\n... - I _J \u2022 \u2022 I \n\n-. \n\n-I \n\n-I - . -J \u2022 \u2022 I \n\n. . . 1 _J \u2022 \u2022 I \n\n-. \n\n~ \n\n-. \n\n~ \n\nJ/ -. \n\n:l\u00a3l':~\"':~:LR' \u00b7:LL\u00b7\u00b7 .~l\u00a3~ \n-. r 1-. \n, , , 'il' '~ \nl~A!il~: TtK~ L. \n'~'kL'LQ'i\u00a3'~'LK\" \n:.. ~ ,:.~ .. : ...:. ,:, ..... ,:.. t?:: .~. ) .. , \n:l C :~ T \n:l \\; \n:L2-,:LL:~::WL:~:LL \n111L \n\n:~ ~ :f \\ t:l ~ tt I \n\n... - , -J \u2022 \u2022 I \n\n. . . . 1 \n\n_ \u2022 \u2022 \u2022 I \n\n. . . 1 oJ \u2022 \u2022 I \n\n... \n\n-I -J \u2022 \u2022 I \n\n... \n\n- I _ \u2022 \u2022 \u2022 I \n\n-I \n\n-I _ \u2022 \u2022 \u2022 I \n\nI \n\n_I \n\n_ I - . \n\n\u2022 \n\n_I _. _. \n\n\u2022 \n\n' . . -. \n\n\u2022 \n\nI \n\n. ' \n\n\u2022 \n\nI \n\nFigure 4: Letters drawn by the reverse TDNN network after 10000 iteration of \nlearning. \n\na gradual reduction of the time resolution. In a reverse TDNN the subsampling \nstarts from the units from the output (which have no subsampling) toward the in(cid:173)\nput. Equivalently, each layer is oversampled when compared to the previous layer. \nThis is illustrated in Figure 3 which shows a small reverse TDNN. The input is \napplied to the 2 units in the lower left. The next layer is unfolded in time two steps \nand has time delay connections toward step zero of the input. The next layer after \nthis is unfolded in time 4 steps (with again time delay connections), and finally the \noutput is completely unfolded in time. The advantage of such an architecture is \nits ability to generate trajectories progressively, starting with the lower frequency \ncomponents at each layer. This parallels recognition TDNN's which extract features \nprogressively. Since the weights are shared between time steps, the network on the \nfigures has only 94 free weights. \n\nWith the reverse TDNN architecture, it was easy to learn the 26 letters of of the \nalphabet. We found that the learning is easier if all the weights are initialized to 0 \nexcept those with the shortest time delay. As a result, the network initially only sees \nits fastest connections. The influence of the remaining connections starts at zero \nand increase as the network learns. The glyphs drawn by the network after 10,000 \ntraining epochs are shown in figure 4. To avoid ambiguity, we give subsampling \nrates with respect to the output, although it would be more natural to mention \noversampling rates with respect to the input. The network has 26 input units, 30 \nhidden units in the first layer subsampled at every 27 time steps, 25 units at the next \nlayer subsampled at every 9 time steps, and 3 output units with no subsampling. \nEvery layer has time delay connections from the previous layer, and is connected \nwith 3 different updates of the previous layer. The time constants were not subject \n\n\fReverse TDNN: An Architecture for Trajectory Generation \n\n587 \n\nto learning and were initialized to 10 for the x and y output units, and to 1 for the \nremaining units. No effort was made to optimize these values. \n\nBig initial time constants prevent the network from making fast variations on the \noutput units and in general slow down the learning process. On the other hand, \nsmall time constants make learning more difficult. The correct strategy is to adapt \nthe time constants to the intrinsic frequencies of the trajectory. With all the time \nconstants equal to one, the network was not able to learn the alphabet (as it was \nthe case in the experiment of the previous section). Good results are obtained with \ntime constants of 10 for the two x-y output units and time constants of 1 for all \nother units. \n\n5 VARIATIONS OF THE ORTDNN \n\nMany variations of the Oversampled Reverse TDNN architecture can be imagined. \nFor example, recurrent connections can be added: connections can go from right to \nleft on figure 3, as long as they go up. Recurrent connections become necessary when \ninformation needs to be stored for an arbitrary long time. Another variation would \nbe to add sensor inputs at various stages of the network, to allow adjustment of the \ntrajectory based on sensor data, either on a global scale (first layers), or locally (last \nlayers). Tasks requiring recurrent ORTDNN's and/or sensor input include dynamic \nrobot control or speech synthesis. \n\nAnother interesting variation is an encoder network consisting of a Subsampled \nTDNN and an Oversmapled Reverse TDNN connected back to back. The Sub(cid:173)\nsampled TDNN encodes the time sequence shown on its input, and the ORTDNN \nreconstructs an time sequence from the output of the TDNN. The main application \nof this network would be the compact encoding of time series. This network can be \ntrained to reproduce its input on its output (auto-encoder), in which case the state \nof the middle layer can be used as a compact code of the input sequence. \n\n6 CONCLUSION \n\nWe have presented a new architecture capable of learning to generate trajectories \nefficiently. The architecture is designed to favor hierarchical representations of tra(cid:173)\njectories in terms of subtasks. \n\nThe experiment shows how the ORTDNN can produce different letters as a function \nof the input. Although this application does not have practical consequences, it \nshows the learning capabilities of the model for generating trajectories. The task \npresented here was particularly difficult because there is no correlation between \nthe patterns. The inputs for an A or a Z only differ on 2 of the 26 input units. \nYet, the network produces totally different trajectories on the output units. This is \npromising since typical neural net application have very correlated patterns which \nare in general much easier to learn. \n\nReferences \n\nBottou, L., Fogelman, F., Blanchet, P., and Lienard, J. S. (1990). Speaker inde-\n\n\f588 \n\nSimard and Le Cun \n\npendent isolated digit recognition: Multilayer perceptron vs Dynamic Time \nWarping. 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IEEE Transactions on Acous(cid:173)\ntics, Speech and Signal Processing, 37:328-339. \n\n\f", "award": [], "sourceid": 577, "authors": [{"given_name": "Patrice", "family_name": "Simard", "institution": null}, {"given_name": "Yann", "family_name": "Le Cun", "institution": null}]}