Rotational Unit of Memory: A Novel Representation - IA de Investigación especializada en el MIT

Rotational Unit of Memory: A Novel Representation
Unit for RNNs with Scalable Applications

Rumen Dangovski*,1, Li Jing*,1, Preslav Nakov2, Mi´co Tatalovi´c1,3, Marin Soljaˇci´c1

*equal contribution
1Instituto de Tecnología de Massachusetts
2Qatar Computing Research Institute, HBKU
3Association of British Science Writers

rumenrd, ljing
}

{

@mit.edu, pnakov@qf.org.qa,

mico, soljacic

@mit.edu

}

{

Abstracto

Stacking long short-term memory (LSTM)
cells or gated recurrent units (GRUs) as part
of a recurrent neural network (RNN) has be-
come a standard approach to solving a number
of tasks ranging from language modeling to
text summarization. Although LSTMs and
GRUs were designed to model
long-range
dependencies more accurately than conven-
tional RNNs, they nevertheless have problems
copying or recalling information from the long
distant past. Aquí, we derive a phase-coded
representation of the memory state, Rotatio-
nal Unit of Memory (RUM), that unifies the
concepts of unitary learning and associative
memory. We show experimentally that RNNs
based on RUMs can solve basic sequential
tasks such as memory copying and mem-
ory recall much better than LSTMs/GRUs.
We further demonstrate that by replacing
LSTM/GRU with RUM units we can apply
neural networks to real-world problems such
as language modeling and text summarization,
yielding results comparable to the state of
the art.

Introducción

An important element of the ongoing neural revo-
lution in Natural Language Processing (NLP) es el
rise of Recurrent Neural Networks (RNNs), cual
have become a standard tool for addressing a num-
ber of tasks ranging from language modeling,
part-of-speech tagging and named entity recogni-
tion to neural machine translation, text summa-
rization, question answering, and building chatbots/
dialog systems.

121

As standard RNNs suffer from exploding/
vanishing gradient problems, alternatives such
as long short-term memory (LSTM) (Hochreiter
and Schmidhuber, 1997) or gated recurrent units
(GRUs) (Cho et al., 2014) have been proposed
and have now become standard.

Sin embargo, LSTMs and GRUs fail to demon-
strate really long-term memory capabilities or
efficient recall on synthetic tasks (ver figura 1).
Cifra 1 shows that when RNN units are fed a long
cadena (p.ej., emojis in Figure 1(a)), they struggle to
represent the input in their memory, which results
in recall or copy mistakes. The origins of these
issues are two-fold: (i) a single hidden state cannot
memorize complicated sequential dynamics and
(ii) the hidden state is not manipulated well,
resulting in information loss. Typically, these are
addressed separately: by using external memory
para (i), and gated mechanisms for (ii).

Aquí, we solve (i) y (ii) jointly by proposing
a novel RNN unit, Rotational Unit of Memory
(RUM),
that manipulates the hidden state by
rotating it in an Euclidean space, resulting in a
better information flow. This remedy to (ii) affects
(i) to the extent that the external memory is less
needed. As a proof of concept, En figura 1(a),
RUM recalls correctly a faraway emoji.

We further show that RUM is fit for real-world
NLP tasks. En figura 1(b), a RUM-based seq2seq
model produces a better summary than what a
standard LSTM-based seq2seq model yields. En
this particular example, LSTM falls into the well-
known trap of repeating information close to the
end, whereas RUM avoids it. De este modo, RUM can
be seen as a more ‘‘well-rounded’’ alternative to
LSTM.

Given the example from Figure 1, we ask the fol-
lowing questions: Does the long-term memory’s

Transacciones de la Asociación de Lingüística Computacional, volumen. 7, páginas. 121–138, 2019. Editor de acciones: Phil Blunsom.
Lote de envío: 8/2018; Lote de revisión: 11/2018; Publicado 4/2019.

2019 Asociación de Lingüística Computacional. Distribuido bajo CC-BY 4.0 licencia.

C
(cid:13)

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then we show promising applications to hard
NLP tasks. Our evaluation of RUM is organized
around a sequence of tests: (1) Pass a synthetic
memory copying test; (2) Pass a synthetic associa-
tive recall test; (3) Show promising performance
for question answering on the bAbI data set;
(4) Improve the state-of-the-art for character-
level language modeling on the Penn Treebank;
(5) Perform effective seq2seq text summariza-
ción, training on the difficult CNN / Daily Mail
summarization corpus.

A lo mejor de nuestro conocimiento, there is no
previous work on RNN units that shows such
promising performance, both theoretical and
practical. Our contributions can be summarized
como sigue: (i) We propose a novel representation
unit for RNNs based on an idea not previously
explored in this context—rotations. (ii) We show
theoretically and experimentally that our unit
models much longer distance dependencies than
LSTM and GRU, and can thus solve tasks such
as memory recall and memory copying much
mejor. (iii) We further demonstrate that RUM
can be used as a replacement for LSTM/GRU in
real-world problems such as language modeling,
question answering, and text summarization,
yielding results comparable to the state of the
art.1

2 Trabajo relacionado

Our work rethinks the concept of gated models.
LSTM and GRU are the most popular such
modelos, and they learn to generate gates—such
as input, reset, and update gates—that modify the
hidden state through element-wise multiplication
and addition. We manipulate the hidden state in
a completely different way: Instead of gates, nosotros
learn directions in the hidden space towards which
we rotate it.

Además, because rotations are orthogonal,
RUM is implicitly orthogonal, meaning that RUM
does not parametrize the orthogonal operation,
but rather extracts it from its own components.
De este modo, RUM is also different from explicitly
orthogonal models such as uRNN, EURNN,
GORU, and all other RNN units that parametrize
their norm-preserving operation (see below).

1Our TensorFlow (Abadi et al., 2015) código, visualizations,
and summaries can be found at http://github.com/
rdangovs/rotational-unit-of-memory.

Cifra 1: RUM vs. LSTM (a) Synthetic sequence
of emojis: A RUM-based RNN recalls the emoji at
posición 1 whereas LSTM does not. (b) Text summa-
rization: A seq2seq model with RUM recalls relevant
information whereas LSTM generates repetitions near
the end.

advantage for synthetic tasks such as copying and
recall translate to improvements for real-world
NLP problems? Can RUM solve issues (i) y
(ii) more efficiently? Does a theoretical advance
improve real-world applications?

We propose RUM as the answer to these
questions via experimental observations and math-
ematical intuition. We combine concepts from
unitary learning and associative memory to uti-
lize the theoretical advantages of rotations, y

122

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0123456789101112? @ 1RUMLSTM(cid:2)(cid:1)(a)(b)Story(abridged)Theraccoonthattopplesyourtrashcanandpillagesyourgardenmayleavemorethanjustamess.Morelikelythannot,italsocontaminatesyouryardwithparasites–mostnotably,raccoonroundwormsbaylisascarisprocyonis(…)ThatistrueinvaryingdegreesthroughoutNorthAmerica,whereurbanraccoonsmayinfectpeoplemorethanpreviouslyassumed.LedbyWeinstein,theUCSBresearcherswonderedifmosthumaninfectionswentundetected…Theirstudy,appearingintheCDCJournalEmergingInfectiousDiseases,foundthat7percentofsurveyedindividualstestedpositiveforraccoonroundwormantibodies.ThatwasnewstoWeinstein,whosaidtheresearcherswouldn’thavebeensurprisedifthey’dfoundnoevidenceofhumaninfection…Over90percentofraccoonsinSantaBarbaraplayhosttothisparasite,whichgrowstoaboutthesizeofaNo.2pencilandcanproduceover100,000eggsperday(…)Sometimestheyreachthebrain,withpotentiallydevastatingconsequences.Thisinfection,termed“baylisascariasis,”killsmice,hasendangeredthealleghenywoodratandhascauseddiseaselikeblindnessorseverebraindamageindozensofpeople,includingatoddlerinSantaBarbarabackin2002.LSTMgeneratedsummary“baylisascariasis,”killsmice,hasendangeredthealleghenywoodratandhascauseddiseaselikeblindnessorsevereconsequences.Thisinfection,termed“baylisascariasis,”killsmice,hasendangeredthealleghenywoodratandhascauseddiseaselikeblindnessorsevereconsequences.Thisinfection,termed“baylisascariasis,”killsmice,hasendangeredthealleghenywoodrat.RUM(nuestro)generatedsummaryUrbanraccoonsmayinfectpeoplemorethanpreviouslyassumed.7percentofsurveyedindividualstestedpositiveforraccoonroundwormantibodies.Over90percentofraccoonsinSantaBarbaraplayhosttothisparasite.

Rotations have fundamental applications in
matemáticas (Artin, 2011; Sala, 2015) and physics
(Sakurai and Napolitano, 2010). In computer
visión, rotational matrices and quaternions con-
tain valuable information and have been used
to estimate object poses (katz, 2001; Shapiro and
Stockman, 2001; Kuipers, 2002). Recientemente, efficient,
accurate and rotationally invariant representation
units have been designed for convolutional neural
redes (Worrall et al., 2017; Cohen et al.,
2018; Weiler et al., 2018). Unlike that work, nosotros
use rotations to design a new RNN unit with
application to NLP, rather than vision.

Unitary learning approaches the problem of
vanishing and exploding gradients, which ob-
struct learning of really long-term dependencies
(Bengio et al., 1994). Teóricamente, using unitary
and orthogonal matrices will keep the norm of
the gradient: the absolute value of their eigen-
values is raised to a high power in the gradient
calculation, but it equals one, so it neither van-
pescados, nor explodes. Arjovsky et al. (2016) (unitary
RNN, or uRNN) and Jing et al. (2017b) (efficient
unitary RNN, or EURNN) used parameteriza-
tions to form the unitary spaces. Wisdom et al.
(2016) applied gradient projection onto a unitary
manifold. Vorontsov et al. (2017) used penalty
terms as a regularization to restrict the matri-
ces to be unitary. Desafortunadamente, such theoretical
approaches struggle to perform outside of the
domain of synthetic tasks, and fail at simple real-
idioma
world tasks such as character-level
modelado (Jing et al., 2017a). To alleviate this
issue, Jing et al. (2017a) combined a unitary pa-
rametrization with gates, thus yielding a gated
orthogonal recurrent unit (GORU), which pro-
vides a ‘‘forgetting mechanism,’’ required by NLP
tareas.

Among pre-existing RNN units, RUM is most
similar to GORU in spirit because both models
transform (de modo significativo) GRU. Nota, sin embargo,
that GORU parametrizes an orthogonal operation
whereas RUM extracts an orthogonal operation in
the form of a rotation. En este sentido, to parallel our
model’s implicit orthogonality to the literature,
RUM is a ‘‘firmware’’ structure rather than a
‘‘learnware’’ structure, as discussed in (Balduzzi
and Ghifary, 2016).

Associative memory modeling provides a large
variety of input encodings in a neural network
for effective pattern recognition (Kohonen, 1974;
Krotov and Hopfield, 2016). It is particularly

123

appealing for RNNs because their memory is
in short supply. RNNs often circumvent this by
using external memory in the form of an attention
mechanism (Bahdanau et al., 2015; Hermann et al.,
2015). Another alternative is the use of neural
Turing machines (Graves et al., 2014, 2016). En
either case, this yields an increase in the size of
the model and makes training harder.

Recent advances in associative memory (Plate,
2003; Danihelka et al., 2016; Ba et al., 2016a;
Zhang and Zhou, 2017) suggest that its updates
can be learned efficiently with backpropagation
through time (Rumelhart et al., 1986). Para examen-
por ejemplo, Zhang and Zhou (2017) used weights that are
dynamically updated by the input sequence. Por
treating the RNN weights as memory determined by
the current input data, a larger memory size is pro-
vided and fewer trainable parameters are required.
Note that none of these methods used rotations
to create the associative memory. The novelty
of our model is that it exploits the simple and
fundamental multiplicative closure of rotations to
generate rotational associative memory for RNNs.
Como resultado, an RNN that uses our RUM units
yields state-of-the-art performance for synthetic
associative recall while using very few parameters.

3 Modelo

Successful RNNs require well-engineered manip-
ulations of the hidden state ht at time step t.
We approach this mathematically, considering ht
as a real vector in an Nh-dimensional Euclidean
espacio, where Nh is the dimension of the ‘‘hidden’’
state RNh. Note that there is an angle between
two vectors in RNh (the cosine of that angle can
be calculated as a normalized dot product ‘‘
'').
Además, we can associate a unique angle to ht
with respect to some reference vector. De este modo, a
hidden state can be characterized by a magnitude,
'', and a phase, es decir., angle with
es decir., L2-norm ‘‘
respect to the reference vector. De este modo, if we devise
a mechanism to generate reference vectors at
any given time step, we would enable rotating
the hidden state with respect to the generated
reference.

(cid:107)

This rethinking of RNh allows us to propose
a powerful
learning representation: Instead of
following the standard way of learning to modify
the norm of ht through multiplication by gates
and self-looping (as in LSTM), we learn to rotate
the hidden state, thereby changing the phase, pero

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Cifra 2: Modelo: (a) RUM’s operation R, which projects and rotates h; (b) the information pipeline in RUM.

preserving the magnitude. The benefits of using
such phase-learning representation are two-fold:
(i) the preserved magnitude yields stable gradients,
which in turn enables really long-term memory,
y (ii) there is always a sequence of rotations that
can bring the current phase to a desired target one,
thus enabling effective recall of information.

In order to achieve this, we need a phase-
learning transformation that is also differentiable.
A simple and efficient approach is to compute the
angle between two special vectors, and then to
update the phase of the hidden state by rotating it
with the computed angle.

We let the RNN generate the special vectors
at time step t (i) by linearly embedding the RNN
RNh,
RNx to an embedded input ˜εt
input xt
y (ii) by obtaining a target memory τ t as a linear
combination of the current input xt (projected in
the hidden space) and the previous history ht
(after a linear transformation).

1
−

∈

→

RNh

The Rotation Operation. We propose a func-
RNh
tion Rotation : RNh
Nh, cual
×
implements this idea. Rotation takes a pair of
column vectors (a, b) and returns the rotational
matrix R from a to b. If a and b have the same
orientación, then R is the identity matrix; otro-
wise, the two vectors form a plane span(a, b). Nuestro
operation projects and rotates in that plane, leaving
everything else intact, as shown in Figure 2(a) para
a = ˜ε and b = τ (por simplicidad, we drop the time
indices).

The computations are as follows. The angle

between two vectors a and b is calculated as

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An orthonormal basis for the plane is (tu, v):

v = (b

−

u = a/
(cid:107)
b)tu)/
(cid:107)

a
(cid:107)
b

(tu

−

b)tu

(cid:107)

We can express the matrix operation R as

−

uu†

−

vv†] + (tu, v) ˜R(i)(tu, v)†

(1)

where the bracketed term is the projection2 and
the second term is the 2D rotation in the plane,
given by the following matrix:

˜R(i) =

cos θ
sin θ

(cid:18)

sin θ
−
cos θ

(cid:19)

≡

Finalmente, we define the rotation operation as fol-
lows: Rotation(a, b)
R. Note that R is
differentiable by construction, and thus it
es
backpropagation-friendly. Además, we imple-
ment Rotation and its action on ht efficiently.
The key consideration is not to compute R explic-
itly. En cambio, we follow Equation (1), which can
be computed in linear memory O(Nh). Asimismo,
the time complexity is O(norte 2

h).

Associative memory. We find that, for some
sequential tasks, it is useful to exploit the multi-
plicative structure of rotations to enable asso-
ciative memory. This is based on the observation
that just like the sum of two real numbers is also a
real number, the product of two rotational matrices
is another rotational matrix.3 Therefore, we use

21 is the identity matrix,

is the transpose of a vector/

matrix and (tu, v) is the concatenation of the two vectors.

†

θ = arccos(a

b/(
(cid:107)

(cid:107)(cid:107)

(cid:107)

))

3This reflects the fact that the set of orthogonal matrices

oh(Nh) forms a group under the multiplication operation.

124

!h#h#$%x#h#+1−+*+!#,#-̃#⊙!#$%!#-̃,h!0h(a)(b)↑2×⊙u#h5#6

a rotation Rt as an additional memory state that
accumulates phase as follows

Rt = (Rt

1)λRotation(˜εt, τ t)
−

(2)

We make the associative memory from Equa-
ción (2) tunable through the parameter λ
,
}
which serves to switch the associative memory off
and on. A lo mejor de nuestro conocimiento, our model
is the first RNN to explore such multiplicative
associative memory.

∈ {

0, 1

Note that even though Rt acts as an additional
memory state, there are no additional parameters
in RUM: The parameters are only within the
Rotation operation. As the same Rotation appears
at each recursive step (2), the parameters are
shared.

∈

The RUM cell. Cifra 2(b) shows a sketch of
the connections in the RUM cell. RUM consists of
RNh that has the same function
an update gate u
as in GRU. Sin embargo, instead of a reset gate, el
RNh. RUM
model learns the memory target τ
∈
RNx into
also learns to embed the input vector x
RNh. Por eso, Rotation encodes
RNh to yield ˜ε
the rotation between the embedded input and the
objetivo, which is accumulated in the associative
Nh (originally initialized
memory unit Rt
×
to the identity matrix). Aquí, λ is a non-negative
integer that is a hyper-parameter of the model.
The orthogonal matrix Rt acts on the state h
to produce an evolved hidden state ˜h. Finalmente,
RUM calculates the new hidden state via u,
just as in GRU. The RUM equations are given
in Algorithm 1. The orthogonal matrix R( ˜εt, t )
conceptually takes the place of a weight kernel
acting on the hidden state in GRU.

RNh

∈

Non-linear activation for RUM. We motivate
the choice of activations using analysis of the
gradient updates. Let the cost function be C. Para
T steps, we compute the partial derivative via the
chain rule:

∂C
∂ht

∂C
∂hT

1
−

∂hk+1
∂hk

∂C
∂hT

1
−

D(k)W †

{

k=t
(cid:89)
1 + Axk + b)
−

k=t
(cid:89)
where D(k) = diag
F (cid:48)(W hk
es el
Jacobian matrix of the pointwise non-linearity f
for a standard vanilla RNN.
For the sake of clarity,

let us consider a
Rk is
simplified version of RUM, where W
a rotation matrix, and let us use spectral norm for
= 1.
matrices. By orthogonality, tenemos

≡

}

W †

(cid:107)

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Cifra 3: Derivatives of popular activations.

(cid:107)

Entonces,

bounded by

the norm of the update
∂C/∂hT

∂C/∂ht
t
1
t
W †
−
−
k=1 (cid:107)
(cid:107)(cid:107)
t
1
∂C/∂hT
.
which simplifies to
−
k=1 (cid:107)
(cid:107)
(cid:107)
D(k)
Por eso, if the norm of
is strictly less than
(cid:107)
uno, we would witness vanishing gradients (para
large T ), which we aim to avoid by choosing a
proper activation.

(cid:107)
D(k)
D(k)

(cid:81)

(cid:107)

Por eso, we compare four well-known activa-
tions f : ReLU, tanh, sigmoid, and softsign.
Cifra 3 shows their derivatives. As long as some
value is positive, the ReLU derivative will be
= 1. This means that ReLU
uno, y por lo tanto
is potentially a good choice. Because RUM is
closer to GRU, which makes the analysis more
complicado, we conduct ablation studies on non-
linear activations and on the importance of the
update gate throughout Section 4.

D(k)

(cid:107)

Time normalization (optional). Sometimes h(cid:48)t
(in Algorithm 1) blows up, Por ejemplo, cuando
using ReLU activation or
for heterogeneous
architectures that use other types of units (p.ej.,
LSTM/GRU) in addition to RUM or perform com-
plex computations based on attention mechanism
or pointers. In such cases, we suggest using L2-
normalization of the hidden state ht to have a fixed
norm η along the time dimension.

We call it time normalization, as we usually
feed mini-batches to the RNN during learning
that have the shape (Nb, NT ), where Nb is the
size of the batch and NT is the length of the
secuencia. Empirically, fixing η to be a small
number stabilizes training, and we found that
values centered around η = 1.0 work well. Este
is an optional component in RUM, as typically
h(cid:48)t does not blow up. En nuestros experimentos, nosotros
only needed it for our character-level language
modelado, which mixes RUM and LSTM units.

125

-4-2240.20.61.

∈

hh, W u(cid:48)

Nh , W τ
×
t , ˜bt
t , bu(cid:48)

RNx
xh, W u(cid:48)
xh ∈
RNx
Nh ; biases bτ
×

Algoritmo 1 Rotational Unit of Memory (RUM)
Nx ; type of
Input: dimensions Nx, Nh, t ; data x
×
cell λ; norm η for time-normalization; non-linear activation
function f .
Initialize: kernels W τ
hh ∈
Nh and ˜Wxh
RNh×
RNh ;
hidden state h0; orthogonal initialization for weights with
gain 1.0.
for t = 1 to T do
1 + bτ
τ t = W τ
hhht
t //memory target
−
1 + bu(cid:48)
tu(cid:48)t = W u(cid:48)
hhht
//update gate
t
−
ut = sigmoid(tu(cid:48)t) //activation of the update gate
˜εt = ˜Wxhxt + ˜bt //embedded input
1)λRotation(˜εt, τ t) //associative memory
Rt = (Rt
−
˜ht = f (˜εt + Rtht
1) //hidden state evolution
−
h(cid:48)t = ut
1 + (1
(cid:12)
ht = ηh(cid:48)t/

˜ht //new state
//normalization N (optional)

xhxt + W τ
xhxt + W u(cid:48)

ut)

(cid:12)

−

∈

−

ht
h(cid:48)t(cid:107)

(cid:107)

end for

4 experimentos

We now describe two kinds of experiments based
(i) on synthetic and (ii) on real-world tasks. El
former test the representational power of RUMs
vs. LSTMs/GRUs, and the latter test whether
RUMs also perform well for real-world NLP
problemas.

4.1 Synthetic Tasks

Copying memory task (A) is a standard testbed
for the RNN’s capability for long-term memory
(Hochreiter and Schmidhuber, 1997; Arjovsky
et al., 2016; Henaff et al., 2016). Aquí, we follow
the experimental set-up in Jing et al. (2017b).

}

{

, i

−

, norte

∈ {

0, 1,

1, norte, norte + 1
}

Datos. The alphabet of the input consists of
, el
symbols
· · ·
first n of which represent data for copying, y
the remaining two forming ‘‘blank’’ and ‘‘marker’’
symbols, respectivamente. En nuestros experimentos, we set
norte = 8 and the data for copying is the first 10
symbols of the input. The RNN model is expected
to output ‘‘blank’’ during T = 500 delay steps and,
after the ‘‘marker’’ appears in the input, to output
(copy) sequentially the first 10 input symbols. El
train/test split is 50,000/500 examples.

Modelos. We test RNNs built using various types
of units: LSTM (Hochreiter and Schmidhuber,
1997), GRU (Cho et al., 2014), uRNN (Wisdom
et al., 2016), EURNN (Jing et al., 2017b),
GORU (Jing et al., 2017a), and RUM (nuestro) con
. We train with
λ
a batch size of 128 and an RMSProp with a 0.9
0.01,
decay rate, and we try learning rates from

1.0, N/A

and η

∈ {

0, 1

}

{

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Cifra 4: Synthetic memory copying results. Shown is
the cross-entropy loss. The number in the name of the
models indicates the size of the hidden state, λ = 1
means tuning the associative memory, and η = N/A
means not using time normalization. Note that the
results for GRU 100 are not visible due to overlap with
GRU 256.

. We found that LSTM and GRU
0.001, 0.0001
}
fail for all learning rates, EURNN is unstable for
large learning rates, and RUM is stable for all
learning rates. De este modo, we use 0.001 for all units
except for EURNN, for which we use 0.0001.

Resultados. En figura 4, we show the cross-entropy
loss for delay time T = 500. Note that LSTM and
GRU hit a predictable baseline of memoryless
estrategia, equivalent to random guessing.4 We can
see that RUM improves over the baseline and
converges to 100% exactitud. For the explicit
unitary models, EURNN and uRNN also solve the
problem in just a few steps, and GORU converges
slightly faster than RUM.

Próximo, we study why RUM units can solve
la tarea, whereas LSTM/GRU units cannot. En
Cifra 4, we also test a RUM model (called RUM(cid:48))
without a flexible target memory and embedded
aporte, eso es, the weight kernels that produce τt
and ˜εt are kept constant. We observe that the
model does not learn well (converges extremely
slowly). This means that learning to rotate the
hidden state by having control over the angles
used for rotations is indeed needed.

Controlling the norm of the hidden state is also
important. The activations of LSTM and GRU
are sigmoid and tanh, respectivamente, and both are
bounded. RUM uses ReLU, which allows larger

4Calculated as follows: C = (M log n)/(T +2M ), dónde
C is cross-entropy, T = 500 is delay time, norte = 8 is the size of
the alphabet, m = 10 is the length of the string to memorize.

126

Modelo

GRU (nuestro)
GORU (nuestro)
EURNN (nuestro)
LSTM (nuestro)
FW-LN (Ba et al., 2016a)
WeiNet (Zhang and Zhou, 2017)
RUM λ = 0 η = N/A (nuestro)
RUM λ = 1 η = 1.0 (nuestro)
RUM λ = 1 η = N/A (nuestro)

Acc.(%)
T = 30/50.
21.5/17.6
21.8/18.9
24.5/18.5
25.6/20.5
100.0/20.8
100.0/100.0
25.0/18.5
100.0/83.7
100.0/100.0

Prms.

14k
13k
4k
17k
9k
22k
13k
13k
13k

Mesa 1: Associative recall results. T is the input
length. Note that line 8 still learns the task completely
for T = 50, but it needs more than 100k training
steps. Además, varying the activations or removing
the update gate does not change the result in the last
line.

hidden states (nevertheless, note that RUM with
the bounded tanh also yields 100% exactitud). Nosotros
observe that, when we remove the normalization,
RUM converges faster compared with using
η = 1.0. Having no time normalization means
larger spikes in the cross-entropy and increased
risk of exploding loss. EURNN and uRNN are
exposed to this, while RUM uses a tunable
reduction of the risk through time normalization.
We also observe the benefits of tuning the
associative rotational memory. En efecto, a RUM
with λ = 1 has a smaller hidden size, Nh = 100,
but it learns much faster than a RUM with λ = 0.
It is possible that the accumulation of phase via
λ = 1 enables faster really long-term memory.

Finalmente, we would like to note that removing the
update gate or using tanh and softsign activations
do not hurt performance.

Associative recall task (B) is another testbed
for long-term memory. We follow the settings in
Ba et al. (2016a) and Zhang and Zhou (2017).

Datos. The sequences for training are random,
and consist of pairs of letters and digits. Nosotros
set the query key to always be a letter. We fix
the size of the letter set to half the length of
the sequence, the digits are from 0 a 9. No
letter is repeated. En particular, the RNN is fed
a sequence of letter–digit pairs followed by the
separation indicator ‘‘??’’ and a query letter (key),
p.ej., ‘‘a1s2d3f4g5??d’’. The RNN is supposed to
output the digit that follows the query key (‘‘d’’ in
this example): It needs to find the query key and
then to output the digit that follows (‘‘3’’ in this
ejemplo). The train/dev/test split is 100k/10k/20k
examples.

Cifra 5: Associative recall study. (a) temperature map
for the weight kernels’ values for a trained model;
(b,C) training evolution of the distribution of cos θ
throughout the sequence of T + 3 = 53 time-steps
(53 numbers in each histogram). For each time step t,
t + 3, we average the values of cos θ across
1
the minibatch dimension and we show the mean.

≤

Modelos. We test LSTM, GRU, GORU, FW-
LN (Ba et al., 2016a), WeiNet (Zhang and Zhou,
2017), and RUM (λ = 1, η = 0). All the models
have the same hidden state Nh = 50 for different
lengths T . We train for 100k epochs with a
tamaño de lote de 128, RMSProp as an optimizer,
and a learning rate of 0.001 (selected using
hyper-parameter search).

Resultados. Mesa 1 shows the results. We can see
that LSTM and GRU are unable to recall the digit
correctly. Even GORU, which learns the copying
tarea, fails to solve the problem. FW-LN, WeiNet,
and RUM can learn the task for T = 30. Para
RUM, it is necessary that λ = 1, as for λ = 0
its performance is similar to that of LSTM and
GORU. WeiNet and RUM are the only known
models that can learn the task for the challenging
50 input characters. Note that RUM yields 100%
accuracy with 40% fewer parameters.

The benefit of

the associative memory is
apparent from the temperature map in Figure 5(a),
where we can see that the weight kernel for the
target memory has a clear diagonal activation. Este
suggests that the model learns how to rotate the
hidden state in the Euclidean space by observing
the sequence encoded in the hidden state. Nota
that none of our baseline models exhibit such a
pattern for the weight kernels.

Cifra 5(b) shows the evolution of the rotational
behavior during the 53 time steps for a model
that does not learn the task. We can see that
cos θ is small and biased towards 0.2. Cifra 5(C)

127

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RUM !=0 %=N/A0.00.20.4(b)10k30k50kRUM !=1 %=N/A(C)0.00.51.04k28k16ktraining steps(a)ModelBPC#ParametersZoneoutLSTM(Kruegeretal.(2016))1.27–RUM2000(nuestro)1.288.9M2⇥RUM1500(nuestro)1.2616.4MHM-LSTM(Chungetal.(2016))1.24–HyperLSTM(Haetal.(2016))1.21914.4MNASCell(Zoph&V.Le(2016))1.21416.3MFS-LSTM-4(Mujikaetal.(2017))1.1936.5MFS-LSTM-2(Mujikaetal.(2017))1.1907.2MFS-RUM-2(nuestro)1.18911.2MTable3:WithFS-RUM-2weachievethestate-of-the-arttestresultonthePennTreebanktask.Additionally,anon-extensivegridsearchforvanillaRNNmodelsyieldscomparableresultstothatofZoneoutLSTM.eachelementonthediagonalactivatesadistinctneuron.Therefore,itseemsthatRUMutilizesthecapacityofthehiddenstatealmostcompletely.Forthisreason,wemightconsiderRUMasanarchitecturethatisclosetothetheoreticaloptimumoftherepresentationalpowerofRNNmodels.Moreover,thediagonalstructureisnottaskspeciﬁc.Forexample,inFigure3(b)weobserveaparticularW(2)hhforthetarget⌧onthePennTreebanktask.Thewayweinterpretthemeaningofthediagonalstructure,combinedwiththeoff-diagonalactivations,isthatprobablytheyencodegrammarandvocabulary,aswellasthelinksbetweenvariouscomponentsoflanguage.!»#(%)!»#(')!##(%)!##(')((')((%))(%))(')diagonal learns text structure (gramática) activate vocabulary, conjugation, etc.…which is effectively a long portion of text…hidden state (neuronas)target memory rotate to align!##(%)kernel for targeta portion of the diagonal, visualized in a horizontal position, has the function to generate a target memory(b)(a)Figure3:ThekernelgeneratingthetargetmemoryforRUMisfollowingadiagonalactivationpattern,whichsigniﬁesthesequentiallearningofthemodel.(a)Atemperaturemapofthevaluesofthevariableswhenthemodelislearned.ThetaskisAssociativeRecall,T=50,andthemodelisRUM,=1,withNh=50andwithouttimenormalization.(b)Aninterpretationofthefunctionofthediagonalandoff-diagonalactivationsofRUM’sWhhkernelonNLPtasks.ThetaskisCharacterLevelPennTreebankandthemodelis=0RUM,Nh=2000,⌘=1.0.SeesectionEforadditionalexamples.5.2THEORETICALANALYSISItisnaturaltoviewtheRotationalUnitofMemoryandmanyotherapproachesusingorthogonalmatricestofallintothecategoryofphase-encodingarchitectures:R=R(✓),where✓isaphaseinformationmatrix.Forinstance,wecanparameterizeanyorthogonalmatrixaccordingtotheEfﬁ-cientUnitaryNeuralNetworks(EUNN,Jingetal.(2017b))architecture:R=QNi=0U0(✓i),whereU0isablockdiagonalmatrixcontainingN/2numbersof2-by-2rotations.Thecomponent✓iisanone-by-(N/2)parametervector.Therefore,therotationalmemoryequationinourmodelcanbe804080104070−5.0−13.0RUM !=1 %=N/Atarget 4and gate 5input 6and hidden state ℎ

shows the evolution of a model with associative
memory (λ = 1) that does learn the task. Nota
that these distributions have a wider range that is
more uniform.

También,

there are one or two cos θ instances
cerca de 1.0 per distribution, eso es, the angle
is close to zero and the hidden state is rotated only
marginally. The distributions in Figure 5(C) producir
more varied representations.

4.2 Real-world NLP Tasks

Question answering (C) is typically done using
neural networks with external memory, but here
we use a vanilla RNN with and without attention.
Datos. We use the bAbI Question Answering
data set (Weston et al., 2016), which consists of
20 subtasks, with 9k/1k/1k examples for train/
dev/test per subtask. We train a separate model
for each subtask. We tokenize the text (at the word
and at the sentence level), and then we concatenate
the story and the question.

For the word level, we embed the words into
dense vectors, and we feed them into the RNN.
the input sequence can be labeled as
Por eso,
1 , . . . , X(s)
X(s)
, where the story has
{
n words and the question has m words.

1 , . . . , X(q)

norte , X(q)

}

metro

}

t
≤
≤

norte , X(q)

For the sentence level, we generate sentence em-
beddings by averaging word embeddings. De este modo,
the input sequence for a story with n sentences
es

1 , . . . , X(s)
X(s)
{
Attention mechanism for sentence level. Usamos
simple dot-product attention (Luong et al., 2015):
h(s)
norte). El
pt
t
t }0
{
≤
≤
context vector c :=
is then passed,
together with the query vector, to a dense layer.
(cid:80)

h(q)
norte := softmax(
·
{
t=0 pth(s)
norte
t

Modelos. We compare uRNN, EURNN, LSTM,
GRU, GORU, and RUM (with η = N/A in
all experiments). The RNN model outputs the
prediction at the end of the question through
a softmax layer. We use a batch size of 32
for all 20 subtasks. We train the model using
Adam optimizer with a learning rate of 0.001
(Kingma and Ba, 2015). All embeddings (palabra
and sentence) are 64-dimensional. For each subset,
we train until convergence on the dev set, sin
other regularization. For testing, we report the
average accuracy over the 20 subtasks.

Resultados. Mesa 2 shows the average accuracy
sobre el 20 bAbI tasks. Without attention, RUM
outperforms LSTM/GRU and all unitary baseline
models by a sizable margin both at the word and

128

Modelo

Acc.(%)

Word Level

1 LSTM (Weston et al., 2016)
2
uRNN (nuestro)
3 EURNN (nuestro)
4 LSTM (nuestro)
5 GRU (nuestro)
6 GORU (Jing et al., 2017a)
7 RUM λ = 0 (nuestro)
8 DNC (Graves et al., 2016)

Sentence Level

9 EUNN/attnEUNN (nuestro)
10 LSTM/attnLSTM (nuestro)
11 GRU/attnGRU (nuestro)
12 GORU/attnGORU (nuestro)
13 RUM/attnRUM λ = 0 (nuestro)
14 RUM/attnRUM λ = 1 (nuestro)
15 RUM/attnRUM λ = 0 w/ tanh (nuestro)
16 MemN2N (Sukhbaatar et al., 2015)
17 GMemN2N (Perez and Liu, 2017)
18 DMN+ (Xiong et al., 2016)
19 EntNet (Henaff et al., 2017)
20 QRN (Seo et al., 2017)

49.2
51.6
52.9
56.0
58.2
60.4
73.2
96.2

66.7/69.5
67.2/80.1
70.4/77.3
71.3/76.4
75.1/74.3
79.0/80.1
70.5/72.9
95.8
96.3
97.2
99.5
99.7

Mesa 2: Question answering results. Accuracy aver-
aged over the 20 bAbI tasks. Using tanh is worse than
ReLU (line 13 vs. 15). RUM 150 λ = 0 without an
update gate drops by 1.7% compared with line 13.

at the sentence level. Además, RUM without
atención (line 14) outperforms all models except
for attnLSTM. Además, LSTM and GRU
benefit
the most from adding attention (líneas
10–11), while the phase-coded models (líneas 9,
12-15) obtain only a small boost in performance
or even a decrease (p.ej., en línea 13). A pesar de
RUM (line 14) shares the best accuracy with
LSTM (line 10), we hypothesize that a ‘‘phase-
inspired’’ attention might further boost RUM’s
performance.5

Language modeling [character-level] (D) es

an important testbed for RNNs (Tumbas, 2013).

Datos. The Penn Treebank (PTB) corpus is a
collection of articles from The Wall Street Journal
(Marcus et al., 1993), with a vocabulary of 10k
palabras (usando 50 different characters). We use a
train/dev/test split of 5.1M/400k/450k tokens, y
we replace rare words with . We feed 150
tokens at a time, and we use a batch size of 128.

Modelos. We incorporate RUM into a recent
high-level model: Fast-Slow RNN (FS-RNN)
(Mujika et al., 2017). The FS-RNN-k architecture

5RUM’s associative memory, Ecuación (2), is similar to
attention because it accumulates phase (es decir., forms a context).
We plan to investigate phase-coded attention in future work.

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consists of two hierarchical layers: one of them is a
‘‘fast’’ layer that connects k RNN cells F1, . . . , Fk
in series; the other is a ‘‘slow’’ layer that consists of
a single RNN cell S. The organization is roughly
como sigue: F1 receives the input from the mini-
batch and feeds its state into S, S feeds its state
into F2, etcétera; finalmente, the output of Fk
is a probability distribution over characters. FS-
RUM-2 uses fast cells (all LSTM) with hidden
tamaño de 700 and a slow cell (RUM) with a hidden
state of size 1000, time normalization η = 1.0,
and λ = 0. We also tried to use associative
memory λ = 1 or to avoid time normalization, pero
we encountered exploding loss at early training
stages. We optimized all hyper-parameters on the
dev set.

Además, we tested FS-EURNN-2, es decir., el
slow cell is EURNN with a hidden size of 2000,
and FS-GORU-2 with a slow cell GORU with a
hidden size of 800 (everything else remains as for
FS-RUM-2). As the learning phases are periodic,
there is no easy regularization for FS-EURNN-2
or FS-GORU-2.

For FS-RNN, we use the hyper-parameter val-
ues suggested in Mujika et al. (2017). We further
use layer normalization (Ba et al., 2016b) on all
estados, on the LSTM gates, on the RUM update
puerta, and on the target memory. We also apply
zoneout (Krueger et al., 2017) to the recurrent
connections, as well as dropout
(Srivastava
et al., 2014). We embed each character into a
128-espacio dimensional (without pre-training).

For training, we use the Adam optimizer with
a learning rate of 0.002, we decay the learning
rate for the last few training epochs, and we apply
gradient clipping with a maximal norm of the
gradients equal to 1.0. Finalmente, we pass the output
through a softmax layer.

For testing, we report bits-per-character (BPC)
loss on the test dataset, which is the cross-entropy
loss but with a binary logarithm.

Our best FS-RUM-2 uses decaying learning
tasa: 180 epochs with a learning rate of 0.002,
entonces 60 epochs with 0.0001, and finally 120 epochs
con 0.00001.

We also test a RUM with η = 1.0, and a two-
layer RUM with η = 0.3. The cell zoneout/hidden
zoneout/dropout probability is 0.5/0.9/0.35 para
FS-RUM-2, y 0.5/0.1/0.65 para
the vanilla
versions. We train for 100 epochs with a 0.002
learning rate. These values were suggested by
Mujika et al. (2017), who used LSTM cells.

Modelo

1 RUM 1400 w/o upd. puerta. (nuestro)
2 RUM 1000 (nuestro)
3 RUM 1000 w/ tanh (nuestro)
4 LSTM (Krueger et al., 2017)
5 LSTM 1000 (nuestro)
6 RUM 1400 (nuestro)
7 RUM 2000 (nuestro)
8 2
RUM 1500 (nuestro)
9 FS-EURNN-2’ (nuestro)
10 FS-GORU-2’ (nuestro)
11 HM-LSTM (Chung et al., 2017)
12 HyperLSTM (Ha et al., 2016)
13 NASCell (Zoph and V. Le, 2017)
14 FS-LSTM-4 (Mujika et al., 2017)
15 FS-LSTM-2 (Mujika et al., 2017)
16 FS-RUM-2 (nuestro)
17 6lyr-QRNN (Merity et al., 2018)
18 3lyr-LSTM (Merity et al., 2018)

BPC
1.326
1.302
1.299
1.270
1.240
1.284
1.280
1.260
1.662
1.559
1.240
1.219
1.214
1.193
1.190
1.189
1.187
1.175

Prms.
2.4METRO
2.4METRO
2.4METRO
–
4.5METRO
4.5METRO
8.9METRO
16.4METRO
14.3METRO
17.0METRO
–
14.4METRO
16.3METRO
6.5METRO
7.2METRO
11.2METRO
13.8METRO
13.8METRO

Mesa 3: Character-level language modeling results.
BPC score on the PTB test split. Using tanh is slightly
better than ReLU (lines 2–3). Removing the update
gate in line 1 is worse than line 2. Phase-inspired reg-
ularization may improve lines 1–3, 6–8, 9–10, y 16.

Resultados.

En mesa 3, we report the BPC loss
for character-level language modeling on PTB.
For the test split, FS-RUM-2 reduces the BPC
for Fast-Slow models by 0.001 points absolute.
Además, we achieved a decrease of 0.002 BPC
points for the validation split using an FS-RUM-2
model with a hidden size of 800 for the slow cell
(RUM) and a hidden size of 1100 for the fast cells
(LSTM). Our results support a conjecture from the
conclusions of Mujika et al. (2017), which states
that models with long-term memory, when used
as the slow cell, may enhance performance.

Text summarization (mi) is the task of reducing
long pieces of text to short summaries without
losing much information. It is one of the most chal-
lenging tasks in NLP (Nenkova and McKeown,
2011), with a number of applications ranging
from question answering to journalism (Tatalovi´c,
2018). Text summarization can be abstractive
(Nallapati et al., 2016), extractive (Nallapati et al.,
2017), or hybrid (See et al., 2017). Avances
in encoder-decoder/seq2seq models (Cho et al.,
2014; Sutskever et al., 2014) established models
based on RNNs as powerful
texto
summarization. Having accumulated knowledge
from the ablation and the preparation tasks, nosotros
test RUM on this hard real-world NLP task.

tools for

Datos. We follow the set-up in See et al. (2017)
and we use the CNN/ Daily Mail corpus (Hermann
et al., 2015; Nallapati et al., 2016), which consist

129

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of news stories with reference summaries. On aver-
edad, hay 781 tokens per story and 56 tokens
per summary. The train/dev/test datasets contain
287,226/13,368/11,490 text–summary pairs.

We further experimented with a new data set,
which we crawled from the Science Daily Web
site, iterating certain patterns of date/time. We suc-
cessfully extracted 60,900 Web pages, each con-
taining a public story about a recent scientific
paper. We extracted the main content, a short sum-
mary, and a title from the HTML page using Beau-
tiful Soup. The input story length is 488.42
219.47,
±
18.60, y
the target summary length is 45.21
±
the title length is 9.35
2.84. En nuestros experimentos,
we set the vocabulary size to 50k.

We defined four tasks on this data: (i) s2s, story
to summary, (ii) sh2s, shuffled story to summary
(we put the paragraphs in the story in a random
orden); (iii) s2t, story to title; y (iv) oods2s, afuera-
of-domain testing for s2s (es decir., training on CNN /
Daily Mail and testing on Science Daily).

Modelos. We use a pointer-generator network
(See et al., 2017), which is a combination of
a seq2seq model (Nallapati et al., 2016) con
atención (Bahdanau et al., 2015) and a pointer
network (Vinyals et al., 2015). We believe that
the pointer-generator network architecture to be a
good testbed for experiments with a new RNN unit
because it enables both abstractive and extractive
summarization.

We adopt the model from See et al. (2017)
as our LEAD baseline. This model uses a
bi-directional LSTM encoder (400 steps) con
attention distribution and an LSTM decoder (100
steps for training and 120 steps for testing),
with all hidden states being 256-dimensional,
and 128-dimensional word embeddings trained
from scratch during training. For training, we use
the cross-entropy loss for the seq2seq model. Para
evaluación, we use ROUGE (Lin and Hovy, 2003).
We also allow the coverage mechanism proposed
in the original paper, which penalizes repetitions
the summaries
and improves the quality of
(marked as ‘‘cov.’’ in Table 4). Following the
original paper, we train LEAD for 270k iterations
and we turn on the coverage for about 3k
iterations at the end to get LEAD cov. Usamos
an Adagrad optimizer with a learning rate of 0.15,
an accumulator value of 0.1, and a batch size of
16. For decoding, we use a beam of size 4.

The only component in LEAD that our proposed
models change is the type of the RNN unit for the

Modelo

1 LEAD (nuestro)
2 decRUM 256 (nuestro)
3 allRUM 360 cov. (nuestro)
4 encRUM 360 cov. (nuestro)
5 decRUM 360 cov. (nuestro)
6 LEAD cov. (nuestro)
7 decRUM 256 cov. (nuestro)
8 (Nallapati et al., 2016)
9 (Nallapati et al., 2017)
10 (See et al., 2017)
11 (See et al., 2017) cov.
12 (Narayan et al., 2018)
13 (Celikyilmaz et al., 2018)
14 (Chen and Bansal, 2018)

ROUGE
2
15.92
16.17
14.69
15.24
16.17
16.86
16.92
13.30
16.20
15.66
17.28
18.20
19.47
18.18

l
33.65
34.07
32.02
33.16
34.23
35.86
36.21
32.65
35.30
33.42
36.38
36.60
37.92
38.79

1
36.89
37.07
35.01
36.34
37.44
39.11
39.54
35.46
39.60
36.44
39.53
40.0
41.69
41.20

L/dR

15 s2s
16 sh2s
17 s2t
18 oods2s

ROUGE (on Science Daily)
2
61.43/57.24
45.24/44.50
10.33/10.56
16.67/22.36

l
65.75/62.03
51.75/51.19
24.81/24.97
26.75/31.11

1
68.83/65.56
56.63/56.13
27.33/27.18
32.91/37.01

}

1,2,l

Mesa 4: Text summarization results. Shown are
ROUGE F-
scores on the test split for the
{
CNN / Daily Mail and the Science Daily datasets.
Some settings are different from ours: lines 8–9 show
results when training and testing on an anonymized
data set, and lines 12–14 use reinforcement learning.
The ROUGE scores have a 95% confidence interval
ranging within
0.25 points absolute. For lines 2
y 7, the maximum decoder steps during testing is
100. In lines 15–18, L/dR stands for LEAD/decRUM.
Replacing ReLU with tanh or removing the update
gate in decRUM line 17 yields a drop in ROUGE
de 0.01/0.09/0.25 y 0.36/0.39/0.42 points absolute,
respectivamente.

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encoder/decoder. Namely, encRUM is a LEAD
with a bidirectional RUM as an encoder (but with
an LSTM decoder), decRUM is LEAD with a
RUM as a decoder (but with a bi-LSTM encoder),
and allRUM is LEAD with all LSTM units
replaced by RUM ones. We train these models
as LEAD, by minimizing the validation cross-
entropy. We found that encRUM and allRUM
take about 100k training steps to converge, mientras
decRUM takes about 270k steps. Entonces, we turn
on coverage training, as advised by See et al.
(2017), and we train for a few thousand steps
. The best ROUGE on dev was
2k,3k,4k,5k,8k
{
achieved for 2k steps, and this is what we used
al final. We did not use time normalization as
training was stable without it. We used the same
hidden sizes for the LSTM, the RUM, y el
mixed models. For the size of the hidden units, nosotros

}

130

256, 360, 400, 512

tried
found that 256 worked best overall.

}

{

on the dev set, and we

Resultados. Mesa 4 shows ROUGE scores for the
CNN / Daily Mail and the Science Daily test splits.
We can see that RUM can easily replace LSTM
in the pointer-generator network. We found that
the best place to use RUM is in the decoder of
the seq2seq model, since decRUM is better than
encRUM and allRUM. En general, we obtained the
best results with decRUM 256 (líneas 2 y 7),
and we observed slight improvements for some
ROUGE variants over previous work (es decir., con
respect to lines 10–11).

We further trained decRUM with coverage for
acerca de 2,000 additional steps, which yielded 0.01
points of increase for ROUGE 1 (but with reduced
ROUGE 2/L). We can conclude that here, como en el
language modeling study (D), a combination of
LSTM and RUM is better than using LSTM-only
or RUM-only seq2seq models.

We conjecture that using RUM in the decoder
is better because the encoder already has an
attention mechanism and thus does not need much
long-term memory, and would better focus on
a more local context (as in LSTM). Sin embargo,
long-term memory is crucial for the decoder as
it has to generate fluent output, and the attention
mechanism cannot help it (es decir., better to use RUM).
This is in line with our attention experiments on
question answering. In future work, we plan to
investigate combinations of LSTM and RUM units
in more detail to identify optimal phase-coded
atención.

Incorporating RUM into the seq2seq model
yields larger gradients, compatible with stable
training. Cifra 6(a) shows the global norm of
the gradients for our baseline models. Because of
the tanh activation, LSTM’s gradients hit the
1.0 baseline even though gradient clipping is
2.0. All RUM-based models have larger global
norm. decRUM 360 sustains a slightly higher
norm than LEAD, which might be beneficial.
Panel 6(b), a consequence of 6(a), demonstrates
that the RUM decoder sustains hidden states of
higher norm throughout training. Panel 6(C) muestra
the contribution of the output at each encoder
step to the gradient updates of the model. Nosotros
observe that an LSTM encoder (in LEAD and
decRUM) yields slightly higher gradient updates
to the model, which is in line with our conjecture
that it is better to use an LSTM encoder. Finalmente,
panel 6(d) shows the gradient updates at each

Cifra 6: Text summarization study on CNN/ Daily
Mail. (a) Global norm of the gradients over time;
(b) Norm of the last hidden state over time; (C) Encoder
gradients of the cost wrt the bi-directional output (400
encoder steps); (d) Decoder gradients of the cost wrt
the decoder output (100 decoder steps). Tenga en cuenta que (C,d)
are evaluated upon convergence, at a specific batch,
and the norms for each time step are averaged across
the batch and the hidden dimension altogether.

decoder step. Although the overall performance
of LEAD and decRUM is similar, we note that the
last few gradient updates from a RUM decoder are
zero, while they are slightly above zero for LSTM.
This happens because the target summaries for a
minibatch are actually shorter than 100 tokens.
Aquí, RUM exhibits an interesting property: Él
identifies that the target summary has ended, y
thus for the subsequent extra steps, our model
stops the gradients from updating the weights. Un
LSTM decoder keeps updating during the extra
steps, which might indicate that it does not identify
the end of the target summary properly.

We also compare our best decRUM 256 modelo
to LEAD on the Science Daily data (lines 15–18).
En mesa 4, lines 15–17, we retrain the models
from scratch. We can see that LEAD has clear
advantage on the easiest task (line 15), cual
generally requires copying the first few sentences
of the Science Daily article.

In line 16, this advantage decreases, as shuffling
the paragraphs makes the task harder. We further
observe that our RUM-based model demonstrates
better performance on ROUGE F-2/L in line 17,
where the task is highly abstractive.

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(a)(b)(C)(d)

Out-of-domain performance.
In line 18, decRUM
256 and LEAD are pretrained on CNN / Daily
Mail (models from lines 1–2), and our RUM-based
model shows clear advantage on all ROUGE met-
rics. We also observe examples that are better
than the ones coming from LEAD (see for exam-
ple the story6 in Figure 1). We hypothesize that
RUM is better on out-of-domain data due to its
associative nature, as can be seen in Equation (2):
At inference, the weight matrix updates for the
hidden state depend explicitly on the current
aporte.

Automating Science Journalism. We further
test decRUM 256 and LEAD on the challenging
task of producing popular summaries for research
artículos. The abundance of such articles online and
the popular coverage of many of them (p.ej., en
Science Daily) provides an opportunity to develop
models for automating science journalism.

The only directly related work7 is that of
Vadapalli et al. (2018), who used research papers
with corresponding popular style blog posts
from Science Daily and phys.org, and aimed
at generating the blog title. In their work, (i)
they fed the paper title and its abstract into a
heuristic function to extract relevant information,
entonces (ii) they fed the output of this function into a
pointer-generator network to produce a candidate
title for the blog post.

Although we also use Science Daily and pointer-
generator networks, we differ from the above work
in a number of aspects. Primero, we focus on gener-
ating highlights, which are longer, more infor-
mative, and more complex than titles. Además,
we feed the model a richer input, que incluye
not only the title and the abstract, but also the full
text of the research paper.8 Finally, we skip (i),

6http://www.sciencedaily.com/releases/

2017/07/170724142035.htm.

7Other summarization work preserved the original scien-

tific style (Teufel and Moens, 2002; Nikolov et al., 2018).

8As the full text for research papers is typically only
available in PDF format (sometimes also in HTML and/or
XML), it is generally hard to convert to text format. De este modo, nosotros
focus on publications by just a few well-known publishers,
which cover a sizable proportion of the research papers
discussed in Science Daily, and for which we developed
analizadores: American Association for the Advancement of
Ciencia (AAAS), Elsevier, Public Library of Science (PLOS),
procedimientos de la Academia Nacional de Ciencias (PNAS),
Saltador, and Wiley. Por último, we ended up with 50,308
full text articles, each paired with a corresponding Science
Daily blog post.

132

Cifra 7: Science Daily-style highlights for the research
paper with DOI 10.1002/smll.201200013.

and in (ii) we encode for 1,000 steps (es decir., aporte
palabras) and we decode for 100 steps. We observed
that reading the first 1,000 words from the research
paper is generally enough to generate a meaningful
Science Daily-style highlight. En general, we encode
much more content from the research paper and
we generate much longer highlights. To the best of
nuestro conocimiento, our model is the only successful
one in the domain of automatic science journalism
that takes such a long input.

Cifra 7 shows some highlights generated by
our models, trained for 35k steps for decRUM
and for 50k steps for LEAD. The highlights
are grammatical, abstractive, and follow the
Science Daily-style of reporting. The pointer-
generator framework also allows for copying
scientific terminology, which allows it to handle
simultaneously domains ranging from computer
ciencia, to physics, to medicine. Curiosamente, el
words cancer and diseases are not mentioned in
the research paper’s title or abstract, not even on
the entire first page; todavía, our models manage to
extract them. See a demo and more examples in
the link at footnote 1.

5 Discusión

RUM vs. GORU. Aquí, we study the energy
landscape of the loss function in order to give some
intuition about why RUM’s choice of rotation is
more appealing than what was used in previous
phase-coded models. Por simplicidad, nosotros sólo
compare to GORU (Jing et al., 2017a) porque

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ScienceDailyreferenceResearchersarecollectingandharvestingenzymeswhilemaintainingtheenzyme’sbioactivity.Thenewmodelsystemmayimpactcancerresearch.LEADgeneratedhighlightScientistshavedevelopedanewmethodthatcouldmakeitpossibletodevelopdrugsandvaccines.Thenewmethodcouldbeusedtodevelopnewdrugstotreatcancerandotherdiseasessuchascancer.decRUMgeneratedhighlightResearchershavedevelopedamethodthatcanbeusedtopredicttheisolationofnanoparticlesinthepresenceofacomplexmixture.Themethod,whichusesnanoparticlestomaptheenzyme,canbeusedtodetectandmonitorenzymes,whichcanbeusedtotreatmetabolicdiseasessuchascancer.

Task Upd. Gate u Best Activations f
ReLU, tanh, sigm.
(A)
cualquier
(B)
ReLU
(C)
ReLU, tanh
(D)
ReLU
(mi)

not needed
not needed
necessary
necessary
necessary

λ
any N/A
N/A
1
N/A
1
1.0
0
N/A
0

Mesa 5: RUM modeling ingredients: Tasks (A–E).

following techniques similar to those in Li et al.
(2018). For each model, we vary the weights
in the orthogonal transformations: the Rotation
operation for RUM, and the phase-coded kernel
in GORU. Cifra 8(a) y 8(C) show a 1D slice
of the energy landscape. Tenga en cuenta que 8(a) has less
local minima than 8(C), which is also seen in
Figures 8(b) y 8(d) for a 2D slice of the energy
landscape.

Note of caution. We should be careful when
using long-term memory RNN units if they are
embedded in more complex networks (not just
vanilla RNNs), such as stacked RNNs or seq2seq
models with attention: Because such networks
use unbounded activations (such as ReLU), el
gradients could blow up in training. This is despite
the unitary mechanism that stabilizes the vanilla
RNN units. Along with the unitary models, RUM
is also susceptible to blow-ups (as LSTM/GRU
son), but it has a tunable mechanism solving this
problema: time normalization.

We end this section with Table 5, which lists
the best ingredients for successful RUM models.

6 Conclusion and Future Work

We have proposed a representation unit for RNNs
that combines properties of unitary learning and
associative memory and enables really long-term
memory modeling. We have further demonstrated
that our model outperforms conventional RNNs
on synthetic and on some real-world NLP tasks.

In future work, we plan to expand the rep-
resentational power of our model by allowing λ
en la ecuación (2) to be not only zero or one, but any
real number.9 Second, we speculate that because

9For a rotational matrix R and a real number λ, nosotros
define the power Rλ through the matrix exponential and the
logarithm of R. Since R is orthogonal, its logarithm is a skew-
symmetric matrix A, and we define Rλ := (eA)λ = eλA.
Note that λA is also skew-symmetric, and thus Rλ is another
orthogonal matrix. For computer implementation, podemos
truncate the series expansion eλA = (cid:80)∞k=0(1/k!)(λA)k at
some late point.

Cifra 8: Energy landscape visualization for our best
RUM (a,b) and GORU (C,d) models on associative
recordar. The first batch from the training split is fixed.
, wδ, wν are randomly
The weight vectors w1, w2, w
chosen instances of the weights used for phase-coding.
Subfigures (a) y (C) show a linear interpolation
by varying α, mientras (b) y (d) visualize a two-
dimensional landscape by varying α and β. All other
weights are fixed, as they do not appear in the rotations.

∗

GORU’s gated mechanism is most similar to
that of RUM, and its orthogonal parametrization,
given by Clements et al. (2016), is similar to
that for the other orthogonal models in Section 2.
j,
Given a batch B =
}
and a model F , the loss L(W., B) is defined as

i, weights W =

{

}

{

j F (W., bj).
In GORU, the weights are defined to be angles

(cid:80)
of rotations, and thus the summand is F (W., bj)
≡
GORU(. . . , porque(Wisconsin), sin(Wisconsin), . . . , bj). The argu-
ments wi of the trigonometric functions are
independent of the batch element bj, and all
summands are in phase. De este modo, the more trigono-
metric functions appear in F (W., bj), the more
local minima we expect to observe in L.

A diferencia de, for RUM we can write F (W., bj)
≡
RUM(. . . , porque(gramo(Wisconsin, bj)), sin(gramo(Wisconsin, bj)), . . . , bj),
where g is the arccos function that was used in
defining the operation Rotation in Section 3.
Because g depends on the input bj, the summands
F (W., bj) are generally out of phase. Como resultado,
L will not be close to periodic, which reduces the
risk of falling into local minima.

We test our intuition by comparing the energy
landscapes of RUM and GORU in Figure 8,

133

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(C)(d)GORU𝛼𝛽𝛼loss1−𝛼𝐰&+𝛼𝐰(𝐰∗+𝛼𝐰*+𝛽𝐰+RUM 𝜆=1(a)(b)

our rotational matrix is a function of the RNN
aporte (rather than being fixed after training, como
in LSTM/GRU), RUM has a lot of potential for
transferir aprendizaje. Finalmente, we would like to explore
novel dataflows for RNN accelerators, which can
run RUM efficiently.

Expresiones de gratitud

We are very grateful to Dan Hogan from Science
Daily for his help, to Daniel Dardani and Matthew
Fucci for their advice, and to Thomas Frerix for
the fruitful discussions.

This work was partially supported by the Army
Research Office through the Institute for Soldier
Nanotechnologies under contract W911NF-18-
2-0048; the National Science Foundation under
grant no. CCF-1640012; and by the Semicon-
ductor Research Corporation under grant no.
2016-EP-2693-B. This research is also supported
in part by the MIT-SenseTime Alliance on
Artificial Intelligence.

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