Proc. of the 15th International Symposium on CMMR, Online, Nov. 15-19, 2021

Hierarchical Predictive Coding
and Interpretable Audio Analysis-Synthesis
André Ofner, Johannes Schleiss and Sebastian Stober
Otto von Guericke University, Magdeburg, Germany{ofner,schleiss,stober}@ovgu.de

Abstract. Humans efficiently extract relevant information from complex auditory
stimuli. Oftentimes, the interpretation of the signal is ambiguous and musical meaning is derived from the subjective context. Predictive processing interpretations ofbrain function describe subjective music experience driven by hierarchical precisionweighted expectations. There is still a lack of efficient and structurally interpretablemachine learning models operating on audio featuring such biological plausibility. We
therefore propose a bio-plausible predictive coding model that analyses auditory signals
in comparison to a continuously updated differentiable generative model. For this,
we discuss and build upon the connections between Infinite Impulse Response filters,
Kalman filters, and the inference in predictive coding with local update rules. Our
results show that such gradient-based predictive coding is useful for classical digital
signal processing applications like audio filtering. We test the model capability on beattracking and audio filtering tasks and conclude by showing how top-down expectations
modulate the activity on lower layers during prediction.
Keywords: Predictive Processing, Machine learning, Digital Signal Processing

1 Introduction
1.1 Audio Processing and Predictive Coding in the Human Brain
Research on human auditory processing has demonstrated that humans are efficient at tracking
stochastic auditory regularities and can even disentangle stationary parts, e.g. fundamental
frequencies, from dynamic transformations, e.g. resonances, in musical events. The predictive
coding (PC) theory is a popular framework in neuroscience that explains how such complex human processing could arise from a relatively simple repeated algorithmic pattern implemented
in neurons, namely the reduction of prediction errors [1, 2]. Recent advances in machinelearning have progressed towards predictive coding models that update simulated neurons witherrors computed local to these neurons, in contrast to the backpropagation through entire neural networks that drive most current deep learning systems [3]. Through the use of local errors
and simple neural operations (e.g. summation or addition) PC networks are plausible modelsof the computations in biological neurons. From an engineering perspective, predictive coding
networks (PCN) with a single layer already deliver useful computations, like the source-filter
separation in Linear Predictive Coding (LPC), a widely used Digital Signal Processing (DSP)
method. To live up to their full potential, PCNs need hierarchical structure. In hierarchical
PCNs hidden layers predict the expected latent states of lower layers. However, there is still a225

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lack of hierarchical and biologically plausible machine learning models that combine the possibility to operate on raw audio with reasonable performance on classical DSP tasks. These tasks
can include audio filtering or extracting musical information, e.g. beat timings, from audio.
1.2 Hierarchical Predictive Coding and Digital Signal Processing
The number of existing studies employing predictive coding to process raw audio is limited andavailable methods are generally difficult to interpret. Moreover, PC models in neuroscience
are generally restricted to simple auditory stimuli or even symbolic inputs [4, 5]. Still, there
are similarities between the structures of Infinite Impulse Response (IIR) filters and recurrent
neural networks (RNN), classes that are already widely used in DSP applications and thosemodels that model human (auditory) cognition more specifically, in particular the Kalman filter
or predictive coding networks. These connections will be discussed in more detail in Section 2.
A major challenge when employing predictive coding networks for engineering tasks isthat they only deliver approximate results during learning and inference. This poses a major
drawback in the context of DSP tasks, where high accuracy is generally required. Furthermore,
it is difficult to design efficiently operating hierarchical PC models, which would have theadvantage of naturally scaling to larger DSP systems with meaningful cognitive interpretations.
To solve these challenges, we resort to the structural similarities between PC models andestablished DSP methods in the next section and then introduce a hierarchical PC model 1.

2 Related Work
The similarity between IIR filters, Kalman filters, RNNs, and predictive coding networks isparticularly apparent when one views these models in their state-space (SSM) form. Figure
1 a) provides an overview of these related classes in state-space form, such as they are usedin tasks typical for each class. Aspects of learned model structure, such as filter coefficients,
are referred to as weights in the context of artificial networks. Generally speaking, ”inference”
refers to employing these given coefficients (i.e. weights) to update hidden representations,while ”learning” refers to the slower process of optimizing weights.
While the signal flow of the model classes is directly comparable, differences arise in
the way inference and learning are addressed in typical tasks. Kalman filters are usually used
for dynamic inference given prior assumptions on the data, resulting in mathematically exact
updates of their latent state. The deterministic class of IIR filters is typically used to apply apreviously designed transfer function to incoming signals, where output signals are a weighted
combination of previously processed signals. Some exceptions, such as differentiable IIR
filters allow to learn weights during application [6]. Kalman filters and predictive coding
networks are typically modeled as probabilistic generative models, keeping track of an inferredlatent state with associated variance (or inverse precision). Both have found applications inmodeling cognitive and neural processes. In contrast to Kalman filters, optimization in predictive coding networks generally addresses state inference and weights learning simultaneously.
Finally, PCNs can include internal predictions of their latent states, i.e. ”top-down” expectations about activities in lower PCN layers [2, 7]. This hierarchical structure is similar,
1

Code is available at github.com/andreofner/APC

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Top-down
prediction

Outputweights
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Observation t

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error tPrediction tb) Signal analysis and synthesis
with autoregressive predictive coding

Fig. 1. a) Comparison of Kalman filters, differentiable IIR filters, and gradient-based predictive coding
networks in state-space form. Blue color indicates variables that are optimized in a typical filtering
application for each model. b) Signal analysis and synthesis with autoregressive predictive coding andlinear activation functions: In the analysis stage, observations at time-step t are mapped to hidden statesusing encoder weights. The learned transition dynamics are then applied to the latent state. Outgoingpredictions for the next timestep t+1 are computed via decoder weights that map from the updated
latent state to the expected sensory input. During synthesis, the prediction error is fed to the model
jointly with the previous prediction.

but not identical, to the multi-layer architecture of deep neural networks, which typicallylack the feedback connections that are inherent to PCNs. More specifically, DNNs can beinterpreted as corresponding to pyramidal dendritic connections in the biological counterpart.
This means that DNNs, possibly with multiple layers, connect adjacent variables in PCN
layers [8]. Finally, existing work on PCN architectures has explored ”dynamical” predictive
coding, where not only the activity of lower layers but also (multiple) temporal derivatives
are modelled [9]. Here, we explore the audio DSP capabilities of single-layer and hierarchical
PCN models interpreted as biologically plausible Neural Kalman filters. This PCN class has
been discussed for single-layer models in [10].
2.1 Autoregressive Signal Filtering with State-Space Models
Signal analysis with autoregressive filters at discrete time-steps t can be described with respect
to a steady state transfer function H(z)
H(z)=

1−

G
k

−j
j=1 aj z

=

G
A(z)

(1)

with input gain G [11, 12]. The parameters aj with 1 ≤ j ≤ k and G of this state transfer
function can be optimized with respect to the prediction error e(x) between predicted signalp(t) and observed signal o(t), also referred to as excitation or residual signal:
e(t)=

k

1
(o(t)− aj o(t−j))
G
j=1

(2)

The SSM of this generalized prediction error filter is updated with the following difference
equation:z[t+1]=A[t]z[t](3)o[t]=C[t]z[t]where z[t] is the state vector at timestep t and the prediction coefficients aj are represented
by weights A and C. All four discussed model classes, despite originating from the different

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fields can be interpreted in prediction error minimizing SSM form. Linear predictive coding(LPC), a widely used DSP tool, draws from this possibility for the design of IIR coefficients.
LPC is typically used for signal compression, particularly for speech coding, by separating
stationary residual signals from imposed resonances [13]. This theoretically allows to analyseand synthesize signals using the same model. However, the efficient algorithms employedin LPC are not directly biologically interpretable and generally do not actually use a SSM
to find the coefficients. From this perspective, our work generalises LPC towards the moregeneral class of hierarchical PCN, where analysis and synthesis use the same model.
RNN and Differentiable IIR Filter Recurrent neural networks, in their simplest form, canbe expressed by the following difference equations [6, 14]:z[t+1]=σz (Wz z[t]+Uz x[t+1]+bz )y[t+1]=σy (Wy z[t+1]+by )(4)with hidden states z, inputs x and outputs y. W and U are trainable weights and b are biases.
Known from previous work is that, in the case where activation functions σ are (non-)linear
and the biases are set to zero, this structure directly resembles a (non-)linear all-pole IIR filterz[t+1]=Wz z[t]+Uz x[t+1]y[t+1]=Wy z[t+1](5)which scales to arbitrary order of transfer functions H(z) (also referred to as the filter order)
and allows to train differentiable IIR filters using the optimization methodology for RNNs
[6]. A useful generalized state space form for such IIR filters isz[t+1]=Az[t]+Bx[t]y[t+1]=Cz[t+1]+Dx[t+1](6)where matrices A,C represent the learnable weights for latent state transition and output
transformation and B,D are weights for input transformations [6].
Kalman Filters The Kalman filter gained large popularity in fields such as engineering,statistics, and neuroscience and filters data points with respect to a probabilistic latent state
and their expected precision. Typically, dynamics and observation models are linear and theobserved noise and the latent states are modeled as Gaussian distributions. Similar to thepreviously discussed model classes, the Kalman filter can be described in SSM form:z[t+1]=Az[t]+Bu[t]+vy[t+1]=Cz[t+1]+w[t](7)with hidden states ht at discrete timesteps t. Correspondingly to the deterministic IIR filter,
the weights of the transition matrix A describe the linear dynamics. The weights of matrix
B and C parameterize the observation model. Weights B transform the control inputs u, i.e.
known inputs to the system and C map from inferred state to the sensory prediction. Finally,v and w are white noise Gaussian processes with mean zero. The Gaussian prior p(zt+1)
and posterior distribution p(zt+1 |y1...t,xt) of the Kalman filter are parameterized by theirsufficient statistics, the mean µ and covariance matrix Σz [10, 15].

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Gradient-Based Predictive Coding Gradient-based predictive coding, as described in has
been applied to an approximation of the exact inference in the Kalman filter [10]. In thesimplest case, without observations or control inputs, we have a state space model of the formz[t+1]=Az[t]y[t+1]=Hz[t+1](8)where A and H are learnable matrices for the state transition dynamics and the observation
model respectively.
Following [10], we define the loss function of the predictive coding filter as:argminµt+1 L=argmaxµt+1 p(yt+1 |zt+1)p(zt+1 |zt)

(9)

In this formulation, weights A and H and the inferred hidden state z (or, more specifically,
its mean µ and variance z parameters) can be updated using gradient descend based on the
precision weighted prediction errors local to the layer [10]:dL=−H T Σz z +Σxx,
dµt+1

dL
=−ΣxxµTt ,dAdL=−y µTt+1
dC

(10)

with sensory prediction errors y =y−Hµt+1 and state prediction errors z =µt+1 −Aµt[10]. Intuitively speaking, this means that each layer optimizes the quality of its signal predictions pyt+1 =Hµt+1 and of its state predictions pµt+1 = Aµt. As this optimization process
happens locally informed and in parallel for each optimized variable, many different possible
outcomes decrease the prediction error. E.g., quickly adapting observation weights H induce
different latent states than a slowly optimized observation model. Similarly, missing accuracy
in the observation model might be compensated by hidden state optimization.
A more general form of the predictive coding SSM includes additional weights for control
inputs u and observed inputs x:
z[t+1]=Az[t]+Bu[t]y[t+1]=Hz[t+1]+Dx[t](11)

In summary, we see that single layer predictive coding models and Kalman filters can be
represented using the same SSM as IIRs and RNNs (excluding nonlinearities), but additionallydifferentiate between control and observed inputs.

3 Hierarchical Predictive Coding of Audio
To create a hierarchy of layers with local computations, we can augment the predictive coding
SSM mentioned in equation 11 with two sets of weights, F and G. These weights modulate
the influence of the layer’s own latent state z in comparison to a top-down prediction of thisstate ztd provided by a higher layer:z[t+1]=F Az[t]+GAztd[t]+Bu[t]y[t+1]=Hz[t+1]+Dx[t](12)and denote the weighted state prediction from current and next higher layer with ẑ =F z + Gztd. In all experiments, we ignore control inputs u, which could receive known229

Proc. of the 15th International Symposium on CMMR, Online, Nov. 15-19, 2021

SINGLE LAYER-TWO LAYERS

-

Error

Error

Observation

Observation

Prediction

Error

Controlencoder

Observation
encoder

Prediction

Observation

Prediction

Prediction

Error

-

Prediction

DecoderStatetransition

Observed-

state

Predicted
state

PrecisionweightingSensory prediction
errorState prediction
error

Error

Fig. 2. Predictive Coding network for hierarchical Kalman filtering: At each timestep t, predictions
yt are generated from a latent state zt using decoder weights that are optimized towards the sensory
prediction error et between observation x and prediction y. Future latent states zt+1 are computed withlearnable transition weights. The transition weights are optimized towards the state prediction error ezt
between predicted state ẑt and the next inferred state zt . Hidden PC layers minimize the prediction error
ezt from a ”top-down” prediction of the state. The hidden state z is optimized towards sensory and stateprediction error et and ezt and creates a balance between outgoing and incoming predictions. Optional
encoders allow to predict with respect to past observations xt−1 or control inputs u.

additional (action) signals and feed past observations xt−1 to the observation encoder for thefiltering task presented in section 4.3.
The state prediction error now includes the additional input and weights:
z =µ[t+1]−F Aµ[t]−GAµtd[t](13)

Figure 2 shows an overview of a single layer predictive coding model and how multiple
layers can be connected through locally informed predictions and prediction error signals.
More precisely speaking, the lowest PC layer directly predicts audio inputs and receives
prediction error et at every timestep. In contrast, hidden PC layers predict the hidden latentstates (”cause units”) of the lower layer and receive state prediction error ezt . Both lowest andhidden PC layers additionally optimize the weights of their transition model that maps fromcurrently inferred state zt to the next state zt +1. We can interpret weights F and G as part ofthe prediction units that produce the optimal state predictions zt+1 given the transition model
A. Finally, the latent state zt +1 is optimized in parallel via gradient descent to minimize thesummed precision weighted prediction error et +ezt local to the respective layer.
We use an overlap-and-add processing approach which is commonly used in DSP, meaning that the PCN processes audio signals in overlapping sequences. For all experiments, thelowest PCN layer processes these sequences sample-by-sample. Hidden layers have identical
update frequencies. We found that sequence sizes between 16 and 2048 frames provide
meaningful results. The hop-length was set to half the sequence length.
3.1 Audio Analysis and Synthesis with Predictive Coding
Assuming purely linear prediction and a well-trained model, using the PCN for audio resynthesis is possible by reverting the process that computes the residual signal at timestep t(i.e. linear prediction error) from the prediction during analysis. Figure 1 b) shows an overview
of the steps for synthesis and analysis given at the lowest layer of a hierarchical predictive

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coding model. While this is not the only possible approach to analyze and synthesize signals
with predictive coding networks, it has the advantage of relatively exactly replicating theapproach taken in LPC. In LPC the coefficients minimizing the squared error during the linear
prediction of the next sample resemble compressed versions of the resonances (typicallyformants in speech coding) and allow the signal to be transmitted with high compression rates
through block-wise filter coefficients and down-sampled residual signals. For linear prediction, this LPC residual signal is equal to the prediction error that arises in (gradient-based)predictive coding.
Assuming linear PCN weights and audio with stationary parts, we expect that resonantparts of the audio are gradually removed from the residual. Added hierarchy and non-linear
activations will affect the meaning of the first layer’s residual signal, e.g. through emerging
attentional processes.

4 Results
4.1 Beat Tracking
In order to quantitatively assess the possibility to extract music information from raw audio
using prediction errors, we resort to a beat tracking task using two datasets: The SMC MIREX
dataset is commonly used for beat tracking evaluation [16]. Our second evaluation is basedon finger tapping recordings in the NMED-T dataset that focuses on electroencephalographic(EEG) recordings during music perception [17]. We choose an approach similar to the predominant local pulse (PLP) method described in Grosche et al. [18] and predict beat timings basedon a local enhancement of a novelty function. The novelty function in [18] is based on spectral
flux, the spectral difference between subsequent Fourier transformed audio inputs. We feed
Fourier transformed audio inputs to the PCN (this being the only place where the PCN inputs
are not audio samples) and use the prediction error from a single layer PCN to compute thenovelty curve. Wherever possible, we use the same FFT parameters as used in Grosche et al.
[18] but do not tune any other hyper parameters. For comparison to other approaches, we report
the F-measure and two continuity-based metrics: CMLt, measuring correctly tracked beats at
the metrical level, and AMLt, which allows variations such as double, half or offbeat variations[19]. All evaluations are based on the mir eval package [20]. Next to the PLP model, we compare our approach to established baselines: A dynamic Bayesian network from [21] and the
dynamic programming approach from [22]. Table 1 shows resulting scores on both datasets.
Table 1. Beat tracking evaluation.
SMC MIREX
F-Score CMLt AMLt
Ellis [22]0.339 0.162 0.315
Grosche [18]0.360 0.071 0.221
Böck - online [21] 0.521 0.363 0.433
PCN (ours)
0.205 0.108 0.201

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NMED-T
F-Score CMLt AMLt
Ellis [22]0.277 0.195 0.473
Grosche [18]0.305 0.037 0.125
Böck - online [21] 0.092 0.105 0.280
PCN (ours)
0.321 0.111 0.295

Proc. of the 15th International Symposium on CMMR, Online, Nov. 15-19, 2021

Interestingly, with respect to the F-Measure, our method outperforms the baselines on
the NMED-T dataset but delivers the worst performance on the SMC dataset. This indicatesa useful performance on genres with salient rhythmical features, as the NMED-T dataset
was designed focusing on Pop songs with clear rhythms. The SMC dataset features manysongs with soft onsets, such as strings, where the novelty function from the prediction error
is not sufficient. We hope that these encouraging results motivate future work with improved
tracking based on predictive coding.

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4.2 Audio Filtering with Top-Down Predictionsa) Repeated audio prediction with 10 state updates b) Repeated audio prediction with 15 state updates
per timestep and 5 updates of the sequence prior. per timestep and 25 updates of the sequence prior.
Fig. 3. a) Repeated prediction of a constant sine wave with single layer (left) and hierarchical PCN with
two layers (right). The hierarchical model learns a top-down state prior for the sequence, while the single
layer model has only local context. When convergence in the lowest layer is not guaranteed, such aswith too few gradient descent steps or with inappropriate initialisation of precision, only the hierarchicalmodel correctly tracks the incoming signal. b) With increased gradient steps for state inference in thelowest layer both single-layer and hierarchical PCN eventually show accurate posterior predictions(green). Predictions from the state prior (blue) improve only for the hierarchical model.

Figure 3 shows examples for repeated block-wise prediction of the same audio input
with a single layer PCN and a hierarchical PCN with two layers for different gradient steps.
In both networks, the inferred state and transition weights of the lowest layer are reset after
each sequence prediction. This means that predictions in the single layer PCN are based on
local information, i.e. the previously seen samples in the sequence. The hierarchical PCN
keeps a top-down prediction of the lower layer’s hidden state, providing refined contextual
information for each prediction. This learnable state prior noticeably leads to a shifted starting
point for the lowest layer in the hierarchical PCN in Fig. 3 a), where the lowest layer has
not enough time to converge properly. When initialised with optimised parameters, bothvariants are able to approximate the target audio to a reasonable degree and the differences in
prediction (and associated prediction errors) are largely restricted to the start of the sequence,as visible in Fig. 3 b). This indicates that minimizing prediction error can be solved through232

Proc. of the 15th International Symposium on CMMR, Online, Nov. 15-19, 2021

online inference in independent trials as well as through the more gradual process of weights
learning when information between trials is carried over. As noticeable in both Fig. 3 a) andb), the learning dynamic of the hierarchical model is significantly more dynamic, since theweighting of the top-down state prior is slightly adapted at each timestep.
The posterior predictions, indicated in Fig. 3 with green lines, show that the lowest PCN
layer does not directly adapt to the top-down prior, but needs some time to tune the remaining weights to this additional source of information. When the top-down prior is correctly
integrated, however, the hierarchical model quickly improves over the single layer model,
especially with parameter initialisation that prevents full convergence of prediction errors in
the lowest layer.
4.3 Replicating Filter Transfer Functions
We tested the possibility to simulate a Butterworth low-pass (LP) filter, which is widely invarious DSP applications. Figure 4 shows input and output audio signals to the targeted LP
filter and the corresponding in and outputs of a PCN. We test PCNs with single and twolayers on a constantly ascending sine wave tone superimposed on constant white noise. BothPCN variants are able to replicate the desired transfer function of the LP filter and show thedesired high frequency content removal.
1.00
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Fig. 4. Replicating an order 2 Butterworth LP filter. LP filter and PCN remove high frequency contents
and have comparable output magnitudes. As the prediction starts with randomized states and without
top-down prior, the prediction error (red) is higher at the sequence start.

5 Conclusion
We presented a gradient-based predictive coding model for audio analysis and synthesis. The
hierarchical model targets biological plausibility through locally informed updates whilestill being efficient and accurate enough to replicate classical DSP tasks like filtering andbeat tracking. We reviewed the similarities between the autoregressive state-space models
underlying predictive coding, IIR filters, recurrent neural networks, and Kalman filtering. The
model provides a basis for future work that could approach more complex DSP applications
or subjectivity in artificial music perception.

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