# Learning Frame Similarity using Siamese networks for Audio-to-Score
Alignment
††thanks: This project has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 765068.

Ruchit Agrawal  Centre for Digital Music
Queen Mary University of London
London, United Kingdom
<EMAIL_ADDRESS>Simon Dixon  Centre for Digital Music
Queen Mary University of London
London, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

Audio-to-score alignment aims at generating an accurate mapping between a
performance audio and the score of a given piece. Standard alignment methods
are based on Dynamic Time Warping (DTW) and employ handcrafted features, which
cannot be adapted to different acoustic conditions. We propose a method to
overcome this limitation using learned frame similarity for audio-to-score
alignment. We focus on offline audio-to-score alignment of piano music.
Experiments on music data from different acoustic conditions demonstrate that
our method achieves higher alignment accuracy than a standard DTW-based method
that uses handcrafted features, and generates robust alignments whilst being
adaptable to different domains at the same time.

###### Index Terms:

Music Information Retrieval, Audio-to-Score Alignment, Siamese networks,
Convolutional Neural Networks, Dynamic Time Warping

##  I Introduction

The significance of neural networks for signal processing was pointed out
early by [1, 2], and their efficacy for Music Information Retrieval (MIR) has
been demonstrated for a variety of tasks like music generation [3], music
transcription [4] as well as music alignment [5]. Audio-to-score alignment is
the task of finding the optimal mapping between a performance and the score
for a given piece of music. Dynamic Time Warping (DTW) [6] has been the de
facto standard for this task, typically incorporating handcrafted features [7,
8, 9]. The primary limitation of handcrafted features lies in their inability
to adapt to different acoustic settings and thereby model real world data in a
robust manner, in addition to not being optimized for the task at hand.

This paper presents a novel method for DTW-based audio-to-score alignment,
which does not depend on handcrafted features, but learns them directly from
the music data at the frame level. We propose learning a frame similarity
matrix using neural networks which is then passed on to a DTW algorithm that
computes the optimal warping path through the matrix, yielding the alignment.
We propose the use of twin Siamese networks [10] each containing a
Convolutional Neural Network (CNN) [11] architecture for learning frame
similarity. The advantage of our method is that it is is efficiently able to
learn meaningful representations for DTW directly from data and is thereby
adaptable to different acoustic settings.

We conduct experiments on piano music using our approach and test its
performance on the Mazurka dataset [12], which contains recordings from
different eras spanning various acoustic conditions; and demonstrate
improvements over _MATCH_ [13], a standard DTW-based method that uses
handcrafted features. We additionally explore two methods to improve the
performance of our baseline models, namely salience representations [14] and
data augmentation.

To the authors’ knowledge, this is the first method to employ learned frame
similarities using Siamese CNNs for audio-to-score alignment. Additionally,
this is the first method to incorporate pitch salience for audio-to-score
alignment to the authors’ knowledge. The rest of the paper is organized as
follows: We describe prior work and our relation to it in Section II. Section
III details our proposed method and model pipeline. The experimentation
conducted and results obtained using our method are described in Section IV.
We present the conclusions of the present research and highlight possible
directions for future work in Section V.

##  II Related Work

Early works on feature learning for Music Information Retrieval (MIR) employ
algorithms like Conditional Random Fields [15] or deep belief networks [16],
whereas recent work in this direction is moving towards the usage of deep
neural networks [17].

Figure 1: Model Pipeline

$conv_{i}$ : |  $i_{th}$ convolution layer |  $pool_{j}$ : |  $j_{th}$ pooling layer
---|---|---|---
$flatten$ : | Flatten layer |  $dense$ : | Fully connected layer

Work specifically on learning features for audio-to-score alignment has
focused on the evaluation of current feature representations [18], learning
features for alignment using a Multi Layer Perceptron [19], and learning a
mapping several common audio representations based on a best-fit criterion
[20]. Recently, transposition-invariant features were proposed for music
alignment [21], however these features while being robust to transposition,
are sensitive to large tempo variations and underperform in such situations.
[22] is a recent work on score following, a task related to audio-to-score
alignment. While they employ reinforcement learning to train a score follower
in real time, we focus on robust offline alignment across various acoustic
conditions using frame similarity learning.

Another direction which sets the context for our work is sound similarity;
approaches to which include capturing music segment similarity using two-
dimensional Fourier-magnitude coefficients [23], similarity network fusion to
combine different frame-level features for hierarchy identification of
repeated sections in music [24], and application of Siamese Neural Networks
for content-based audio retrieval [25]. The closest work to ours which employs
the notion of learned sound similarity for music alignment is [19], to the
authors’ knowledge. While they use a Multi-Layer Perceptron to compute if two
frames are the same or not, we compute frame similarity using Siamese CNNs. In
addition to using an enhanced framework which is suitable for the similarity
detection task, our work differs from them in that we also compute the extent
of similarity in the form of non-binary distances and use this distance (or
dissimilarity) matrix further for alignment. We additionally employ deep
salience representations, which prove to be an effective method to improve
alignment accuracy over our baseline models.

##  III Proposed Methodology

We propose a novel method for DTW-based audio-to-score alignment that uses
Siamese neural networks. We additionally employ deep salience representations
[14] to improve model performance in data-scarce conditions. We describe the
method in detail in the subsequent subsections.

###  III-A Siamese Convolutional Neural Networks

The standard feature representation choice for music alignment is a time-
chroma representation [26] generated from the log-frequency spectrogram, which
is not trainable on real data, and thereby not adaptable to different acoustic
settings. We override the feature engineering step and focus on learning frame
similarity using Convolutional Neural Networks (CNNs), since they can jointly
optimize the representation of input data conditioned on the similarity
measure being used. We employ a Siamese Convolutional Neural Network, a class
of neural network architectures that contains two or more identical
subnetworks [10] for this task.

We train a Siamese CNN, akin to that prototyped in [27], to compute a frame
similarity matrix $S_{m}$ to be fed to DTW to generate alignment. Figure 1
gives an overview of our model pipeline. In order to keep the modality
constant, we first convert the MIDI files to audio through FluidSynth [28]
using piano soundfonts. The two audio inputs are converted to a low-level
spectral representation using a Short Time Fourier Transform, with a hop size
of 23 ms and a hamming window of size 46 ms. Our training data contains
synchronized audio and MIDI files, so it is straightforward to extract
matching frame pairs. For each matching pair, we randomly select a non-
matching pair (using MIDI-information) in order to have a balanced training
set. The inputs to the Siamese network are labelled frame pairs from the
performance audio and the synthesized MIDI respectively. We employ the
contrastive loss function [29] while training our models. We choose this
formulation over a standard classification loss function like cross entropy
since our objective is to differentiate between two audio frames. Let
$X=(X_{1},X_{2})$ be the pair of inputs $X_{1}$ and $X_{2}$, $W$ be the set of
parameters to be learnt and $Y$ be the target binary label ($Y$ = 0 if they
match and 1 if otherwise). Task-specific loss functions have shown promising
results in the fields of image processing and natural language processing [30,
31]. The contrastive loss function for each tuple is computed as follows:

$L(W,X,Y)=(1-Y)\frac{1}{2}(D_{W})^{2}+(Y)\frac{1}{2}\\{max(0,m-D_{W})\\}^{2}$
(1)

where $m$ is the margin for dissimilarity and $D_{W}$ is the Euclidean
Distance between the outputs of the subnetworks. Pairs with dissimilarity
greater than $m$ do not contribute to the loss function. More formally,
$D_{W}$ can be expressed as follows:

$D_{W}(X)=\sqrt{\\{G_{W}(X_{1})-G_{W}(X_{2})\\}^{2}}$ (2)

where $G_{W}$ is the output of each twin subnetwork for the inputs $X_{1}$ and
$X_{2}$. Since it is a distance-based loss, it tries to ensure that
semantically similar examples are embedded close to each other, which is a
desirable trait for extracting alignments.

The Siamese network thus learns to classify the sample pairs as similar or
dissimilar. This is done for each audio frame pair and the similarity matrix
thus generated is then passed on to a DTW-based algorithm to generate the
alignment path. DTW generates an alignment between two sequences $A$=
$(a_{1},a_{2},...,a_{m})$ and $B$ = $(b_{1},b_{2},...,b_{n})$ by comparing
them using a local cost function, at each point, with the goal of minimizing
the overall cost. The path which yields this minimum overall cost is then the
optimal alignment between the two sequences. Formally, it can be represented
as follows:

$D(i,j)=d(i,j)+min\begin{cases}D(i,j-1)\\\ D(i-1,j)\\\ D(i-1,j-1)\end{cases}$
(3)

where $d(i,j)$ is the distance measure (local cost) between points $a_{i}$ and
$b_{j}$; and $D(i,j)$ is the total cost for the path which generates the
optimal alignment between the sequences $A_{1..i}$ and $B_{1..j}$. We employ
Euclidean distance as our distance measure and the DTW framework from [32] to
compute the warping paths.

###  III-B Deep Salience Representations

Figure 2: Salience representations to address data sparsity

We employ deep salience representations [14] for effective training of our
models. These are time-frequency representations aimed at estimating the
likelihood of a pitch being present in the audio. Figure 2 shows an example of
a salience representation. The primary motivation behind using such a
representation is that it de-emphasizes non-pitched content and emphasizes
harmonic content, thereby aiding training in data-scarce conditions. We employ
the model proposed by [14], trained to learn a series of convolutional
filters, constraining the target salience representation to have values
between 0 and 1, with larger values corresponding to time-frequency bins where
fundamental frequencies are present. The model is trained to minimize the
cross entropy loss as follows:

$L(y,\hat{y})=-ylog(\hat{y})-(1-y)log(1-\hat{y})$ (4)

where both $y$ and $\hat{y}$ are continuous values between 0 and 1.

We compare the performance obtained using salience representations with that
obtained using the Short-Time Fourier Transform (STFT) and Constant-Q
Transform (CQT) of the raw audios. We employ these input representations for
comparative purposes. We employ a hop size of 23 ms and a hamming window of
size 46 ms. We employ a CQT with 24 bins per octave, with the first bin
corresponding to frequency 65.4 Hz (midi note C2).

##  IV Experiments and Results

###  IV-A Experimental Setup

We employ the MAPS database [33], the Saarland database [34] and the Mazurka
dataset [12] for our experiments. From the original MAPS database, which
contains synthesized MIDI-aligned audio for a range of acoustic settings, we
select the subset _MUS_ containing complete pieces of piano music, and append
it to the Saarland database. We split the resultant database comprising 288
recordings randomly into sets of 230 and 58 recordings. These sets form our
training and validation sets respectively. We test the performance of our
models on the Mazurka dataset [12], which contains recordings of Chopin’s
Mazurkas dating from 1902 to the early 2000s, thereby spanning across various
acoustic settings. This dataset contains annotations of beat times for five
Mazurka pieces. The alignment error for these pieces has a standard deviation
of 11 ms.

TABLE I: Architecture of our model Type of layer | Input size | Kernels | Kernel size
---|---|---|---
Convolution | $128*128*3$ | 64 | $5*5$
Max-Pooling | $128*128*64$ | 1 | $2*2$
Convolution | $64*64*64$ | 128 | $5*5$
Max-Pooling | $64*64*128$ | 1 | $2*2$
Convolution | $32*32*128$ | 256 | $3*3$
Max-Pooling | $32*32*256$ | 1 | $2*2$
Convolution | $16*16*256$ | 512 | $3*3$
Flatten | $16*16*512$ | - | -
Fully Connected | $131072$ | - | -
TABLE II: Results of our models Model | Binary Matrix | Distance Matrix
---|---|---
$<$25ms |  $<$50ms |  $<$100ms |  $<$200ms |  $<$25ms |  $<$50ms |  $<$100ms |  $<$200ms
$MATCH$[13] | - | - | - | - | 64.8 | 72.1 | 77.6 | 83.7
$DTW_{Chroma}$ | - | - | - | - | 62.9 | 70.5 | 76.3 | 82.4
$MLP_{Semigram}$[19] | 63.8 | 69.5 | 77.2 | 83.4 | - | - | - | -
$\mathit{SCNN_{STFT}}$ | 65.6 | 71.9 | 78.1 | 84.8 | 67.2 | 73.4 | 78.7 | 85.6
$\mathit{SCNN_{CQT}}$ | 66.4 | 73.1 | 78.7 | 85.3 | 68.1 | 74.8 | 80.1 | 86.7
$\mathit{SCNN_{Chroma}}$ | 67.1 | 74.6 | 79.2 | 86.1 | 69.4 | 75.1 | 80.7 | 87.2
$\mathit{SCNN_{Sal}}$ | 68.2 | 75.3 | 81.4 | 87.8 | 70.3 | 76.7 | 82.1 | 88.4
$\mathit{SCNN_{CQT+DA}}$ | 67.9 | 74.4 | 80.8 | 86.7 | 69.6 | 75.4 | 81.6 | 87.9
$\mathit{SCNN_{Sal+DA}}$ | 69.4 | 76.4 | 81.2 | 87.5 | 71.7 | 78.2 | 83.3 | 90.1

Our Siamese model has four convolutional layers of varying dimensionality
followed by a fully connected layer to generate the similarity output. The
outputs of each layer are passed through rectified linear units in order to
add non-linearity, followed by batch normalization before being passed as
inputs to the next layer. The detailed architecture of our model is given in
Table I.

We conduct experiments using two different mechanisms for computing the
similarity matrix $S_{m}$:

  * •

Using binary labels: We directly employ the outputs of the Siamese CNN,
whereby 0 and 1 correspond to similar and dissimilar pairs respectively.

  * •

Using distances: We employ the distance $D_{W}$ computed as part of the loss,
which directly corresponds to the dissimilarity between the two inputs.

We generate an alignment path through this matrix using DTW, through a readily
available implementation in Python [32]. For our Siamese models trained
without data augmentation, the naming convention we employ is
$\mathit{SCNN_{x}}$, where $x$ is the feature representation used during
training. We also report results obtained using data augmentation. We generate
20% additional training samples by employing a random pitch shift of up to
$\pm 30\text{\,}\mathrm{c}$ents, using librosa [35]. These models are named
$\mathit{SCNN_{CQT+DA}}$ and $\mathit{SCNN_{Sal+DA}}$ for the CQT and the
salience representations respectively.

###  IV-B Results and Discussion

We compare the performance of our models with _MATCH_ [13]; a DTW algorithm
using Chroma features [26]; and the Multi-Layer Perceptron Model proposed by
[19] ($MLP_{Semigram}$). We compute the error $e_{i}$ = ${t_{i}}^{e}$ \-
${t_{i}}^{r}$, defined as the time difference between the alignment positions
of corresponding events in the reference ${t_{i}}^{r}$ and the estimated
alignment time ${t_{i}}^{e}$ for score event $i$. We show results for accuracy
in percentage for events which are aligned within an error of up to 25 ms,
50ms, 100ms and 200ms respectively. The results obtained by our models are
given in Table II.

Our models outperform DTW-based algorithms that employ handcrafted features as
well as an MLP framework which learns binary similarity labels (Table II, rows
1-5). The CQT representation ($\mathit{SCNN_{CQT}}$) yields better results
than the STFT representation ($\mathit{SCNN_{STFT}}$), we argue that this is
due to the nature of the CQT, which is a more musically meaningful
representation. Our Siamese model trained using the Chroma representation
($\mathit{SCNN_{Chroma}}$) outperforms the DTW-based method using the same
representation ($DTW_{Chroma}$), suggesting that frame similarity learnt from
real data is effective at generating robust alignment. Additionally, we
observe the trend that the models trained using a non-binary distance matrix
outperform those trained on binary matrices (Table II, columns 6-9). We
speculate that thresholding the similarity into binary labels discards
potentially useful information and the distances facilitate the DTW algorithm
to take better long-term decisions. Both salience representations
($\mathit{SCNN_{Sal}}$) and data augmentation ($\mathit{SCNN_{DA}}$) prove to
be effective to improve the performance of our model over
$\mathit{SCNN_{CQT}}$, with salience representations contributing to greater
improvements. We posit that using salience representations makes it easier for
the model to learn meaningful features from the input representations, since
it emphasizes pitched content. Improvements using data augmentation can be
attributed to the fact that pianos are not always tuned to $A=440Hz$ in the
real world, and often the relative intervals are also not tuned perfectly,
hence comparison with MIDI files in such cases might lead to false negatives.
Data augmentation ensures that the disparity between our training and test
conditions is minimized by simulating more real-world like conditions in our
training data. A combination of distance matrix, salience representations and
data augmentation yields the best results ($\mathit{SCNN_{Sal+DA}}$), as can
be seen from Table II, row 8, columns 6-9.

Our results demonstrate that frame similarity learning using Siamese neural
networks is a promising method for audio-to-score alignment. The principal
advantage of this approach over traditional feature choices (like chroma
features or MFCCs) is the ability to learn directly from data, which provides
higher relevance and adaptability. Both the Siamese network and the pitch
salience network are trainable, and thereby adaptable to real world
conditions. We plan to explore domain adaptation of our models in the future.
A limitation of our method is that it cannot handle structural changes, since
DTW generates a monotonically increasing warping path. This could potentially
be mitigated by employing an enhanced DTW framework like jump-DTW [36]
alongside our Siamese model.

##  V Conclusion and Future Work

We presented a novel method for offline audio-to-score alignment using learned
similarities via a Siamese convolutional network architecture. We demonstrated
that our approach is capable of generating robust alignments for piano music
across various acoustic conditions. Our models outperform traditional methods
based on Dynamic Time Warping that rely on handcrafted features, as well as a
Multi Layer Perceptron model which learns binary similarity between audio
frames. We also demonstrated that salience representations and data
augmentation are effective techniques to improve alignment accuracy. In the
future we plan to incorporate attention into the convolutional models to aid
training and improve performance. We would also like to explore other model
architectures and work on learning the features as well as the alignments in a
completely end-to-end manner.

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p)}{np}}+\sqrt{r}\kappa\bigg{(}\frac{1}{n}+\sqrt{\frac{1}{pn}}+\frac{1}{p}\sqrt{\frac{\log(n\vee
p)}{n}}\bigg{)}$ $\displaystyle\qquad+\sqrt{r}\bigg{(}\frac{\mathcal{E}^{{\sf
PCA}}}{\lambda_{\min}}\bigg{)}^{2}\bigg{(}\frac{1}{p}+\frac{1}{n}+\sqrt{\frac{1}{pn}}+\frac{1}{p}\sqrt{\frac{\log(n\vee
p)}{n}}\bigg{)}+\sqrt{r}\frac{\mathcal{E}^{{\sf PCA}}}{\lambda_{\min}p}$
$\displaystyle\lesssim\sqrt{r}\kappa\frac{\log^{4}(n\vee
p)}{n}+\frac{\sqrt{r}}{p}\frac{\mathcal{E}^{{\sf PCA}}}{\lambda_{\min}},$

with probability at least $1-O((n\vee p)^{-10})$, where we have used the
assumption that $p>n/\log^{4}(n\vee p)$. This proves the first assertion.

We now prove the second assertion, where we assume that $p\leq
n/\log^{4}(n\vee p)$, which in particular implies $p\leq n$. First, by the
Poincare Separation Theorem (Corollary 4.3.37 of Horn and Johnson (2012)) it
holds that

$\displaystyle\widehat{\lambda}_{r+1}\leq\lambda_{r}\bigg{(}\bm{U}_{\perp}^{\top}\frac{\bm{XX}^{\top}}{n}\bm{U}_{\perp}\bigg{)}=\sigma^{2}\lambda_{r}\bigg{(}\frac{\bm{U}_{\perp}^{\top}\bm{YY}^{\top}\bm{U}_{\perp}}{n}\bigg{)}.$

Note that $\bm{U}_{\perp}^{\top}\bm{Y}$ is a $p-r\times n$-dimensional matrix
of independent standard Gaussian random variables. Therefore, the covariance
concentration inequality (e.g., Lemma 8.7) implies that with probability at
least $1-O((n\vee p)^{-10})$,

$\displaystyle\bigg{\|}\frac{\bm{U}_{\perp}^{\top}\bm{YY}^{\top}\bm{U}_{\perp}}{n}-\bm{I}_{p-r}\bigg{\|}$
$\displaystyle\lesssim\sqrt{\frac{p}{n}}+\sqrt{\frac{\log(p\vee n)}{n}},$ (66)

since $p\leq n$. Therefore, Weyl’s inequality implies that

$\displaystyle\sigma^{2}\lambda_{r}\bigg{(}\frac{\bm{U}_{\perp}^{\top}\bm{YY}^{\top}\bm{U}_{\perp}}{n}\bigg{)}$
$\displaystyle\leq\sigma^{2}+O\bigg{(}\sigma^{2}\sqrt{\frac{p}{n}}+\sigma^{2}\sqrt{\frac{\log(p\vee
n)}{n}}\bigg{)}.$

Next, we have that

$\displaystyle\widehat{\lambda}_{r+1}$
$\displaystyle=\lambda_{r+1}\bigg{(}\bm{\Sigma}^{1/2}\frac{\bm{YY}^{\top}}{n}\bm{\Sigma}^{1/2}\bigg{)}$
$\displaystyle\geq\sigma^{2}\lambda_{r+1}\bigg{(}\frac{\bm{Y}\bm{Y}^{\top}}{n}\bigg{)}$
$\displaystyle\geq\sigma^{2}-O\bigg{(}\sigma^{2}\sqrt{\frac{p}{n}}+\sigma^{2}\sqrt{\frac{\log(p\vee
n)}{n}}\bigg{)},$

where we have applied Weyl’s inequality and the bound (66). Consequently,
combining the upper and lower bound, we have that

$\displaystyle\big{|}\frac{\widehat{\lambda}_{r+1}}{\sigma^{2}}-1\big{|}$
$\displaystyle\lesssim\sqrt{\frac{p}{n}}+\sqrt{\frac{\log(p\vee n)}{n}},$

which holds with probability at least $1-O((n\vee p)^{-10})$, which completes
the proof. ∎

###  E.9 Proof of Lemma C.10

###### Proof.

First by Lemma 8.12 it holds that

$\displaystyle\sqrt{1+\widehat{b_{k}^{{\sf PCA}}}}=\sqrt{1+b_{k}^{{\sf
PCA}}}+O\bigg{(}\frac{\sigma^{2}p}{\lambda_{\min}n}\bigg{(}\sqrt{r}\kappa\bigg{(}\frac{1}{n}+\sqrt{\frac{\log(n\vee
p)}{np}}\bigg{)}+\frac{\sqrt{r}}{p}\frac{\mathcal{E}^{{\sf
PCA}}}{\lambda_{\min}}\bigg{)},$

for $p>n$, and where the $O(\cdot)$ term above is zero for $p\leq n$.
Furthermore, by Lemma 8.8, with probability at least $1-O((n\vee p)^{-10})$ it
holds that

$\displaystyle\bigg{|}1-\sqrt{1+b_{k}^{{\sf
PCA}}}\bm{u}_{k}^{\top}\bm{\widehat{u}}_{k}\bigg{|}$
$\displaystyle\lesssim\frac{(\lambda_{\max}+\sigma^{2})(\lambda_{k}+\sigma^{2})r\log(n\vee
p)}{\Delta_{k}^{2}n}$
$\displaystyle\quad+\frac{\sigma(\lambda_{k}+\sigma^{2})^{1/2}}{\lambda_{k}\sqrt{n}}\bigg{(}\frac{\sigma\kappa\sqrt{r}\log^{2}(n\vee
p)}{\lambda_{\min}^{1/2}}\sqrt{\frac{p}{n}}+\frac{\sigma^{2}\sqrt{r\kappa}\log^{2}(n\vee
p)}{\lambda_{\min}}\sqrt{\frac{p}{n}}\bigg{)}.$

Therefore, if $p>n$,

$\displaystyle\bigg{|}\sqrt{1+\widehat{b_{k}^{{\sf
PCA}}}}\bm{u}_{k}^{\top}\bm{\widehat{u}}_{k}-1\bigg{|}$
$\displaystyle\lesssim\frac{(\lambda_{\max}+\sigma^{2})(\lambda_{k}+\sigma^{2})r\log(n\vee
p)}{\Delta_{k}^{2}n}$
$\displaystyle\quad+\frac{\sigma(\lambda_{k}+\sigma^{2})^{1/2}}{\lambda_{k}\sqrt{n}}\bigg{(}\frac{\sigma\kappa\sqrt{r}\log^{2}(n\vee
p)}{\lambda_{\min}^{1/2}}\sqrt{\frac{p}{n}}+\frac{\sigma^{2}\sqrt{r\kappa}\log^{2}(n\vee
p)}{\lambda_{\min}}\sqrt{\frac{p}{n}}\bigg{)}$
$\displaystyle\quad+\frac{\sigma^{2}p}{\lambda_{\min}n}\bigg{(}\sqrt{r}\kappa\bigg{(}\frac{1}{n}+\sqrt{\frac{\log(n\vee
p)}{np}}\bigg{)}+\frac{\sqrt{r}}{p}\frac{\mathcal{E}^{{\sf
PCA}}}{\lambda_{\min}}$
$\displaystyle\lesssim\frac{(\lambda_{\max}+\sigma^{2})(\lambda_{k}+\sigma^{2})r\log(n\vee
p)}{\Delta_{k}^{2}n}$
$\displaystyle\quad+\bigg{(}\frac{\sigma}{\lambda_{\min}^{1/2}\sqrt{n}}+\frac{\sigma^{2}}{\lambda_{\min}\sqrt{n}}\bigg{)}\bigg{(}\frac{\sigma\kappa\sqrt{r}\log^{2}(n\vee
p)}{\lambda_{\min}^{1/2}}\sqrt{\frac{p}{n}}+\frac{\sigma^{2}\sqrt{r\kappa}\log^{2}(n\vee
p)}{\lambda_{\min}}\sqrt{\frac{p}{n}}\bigg{)}$
$\displaystyle\quad+\frac{\sigma^{2}p}{\lambda_{\min}n}\bigg{(}\sqrt{r}\kappa\bigg{(}\frac{1}{n}+\sqrt{\frac{\log(n\vee
p)}{np}}\bigg{)}+\frac{\sqrt{r}}{p}\frac{\mathcal{E}^{{\sf
PCA}}}{\lambda_{\min}}$ $\displaystyle=\mathcal{E}_{b},$

where the same bound holds with $p\leq n$ by observing that the final term is
zero. By the analysis leading to the proof of Theorem 3.3, we have that with
probability at least $1-O((n\vee p)^{-8})$,

$\displaystyle\big{|}\bm{a}^{\top}\bm{\widehat{u}}_{k}\sqrt{1+\widehat{b_{k}^{{\sf
PCA}}}}-\bm{a}^{\top}\bm{u}_{k}\big{|}$
$\displaystyle\leq\bigg{|}\big{(}\bm{a}^{\top}\bm{\widehat{u}}_{k}-\bm{a}^{\top}\bm{u}_{k}\bm{u}_{k}^{\top}\bm{\widehat{u}}_{k}\big{)}\sqrt{1+\widehat{b_{k}^{{\sf
PCA}}}}\bigg{|}+\big{|}\bm{a}^{\top}\bm{u}_{k}\big{|}\big{|}1-\bm{u}_{k}\bm{u}_{k}^{\top}\bm{\widehat{u}}_{k}\sqrt{1+\widehat{b_{k}^{{\sf
PCA}}}}\big{|}$ $\displaystyle\leq\bigg{|}\bigg{(}\sum_{l\neq
k}\frac{\bm{u}_{k}^{\top}\big{(}\bm{\widehat{\Sigma}}-\bm{\Sigma}\big{)}\bm{u}_{l}}{\lambda_{k}-\lambda_{l}}(\bm{u}_{l}^{\top}\bm{a})+s^{{\sf
PCA}}_{\bm{a},k}\times{\sf ErrPCA}\bigg{)}\sqrt{1+\widehat{b_{k}^{{\sf
PCA}}}}\bigg{|}$
$\displaystyle\quad+\big{|}\bm{a}^{\top}\bm{u}_{k}\big{|}\big{|}1-\bm{u}_{k}\bm{u}_{k}^{\top}\bm{\widehat{u}}_{k}\sqrt{1+\widehat{b_{k}^{{\sf
PCA}}}}\big{|}$ $\displaystyle\lesssim s_{\bm{a},k}^{{\sf
PCA}}\bigg{(}\log(n\vee p)+{\sf
ErrPCA}\bigg{)}+|\bm{a}^{\top}\bm{u}_{k}|\mathcal{E}_{b}$
$\displaystyle\lesssim s_{\bm{a},j}^{{\sf PCA}}\frac{s_{\bm{a},k}^{{\sf
PCA}}}{s_{\bm{a},j}^{{\sf PCA}}}\log(n\vee
p)+|\bm{a}^{\top}\bm{u}_{k}|\mathcal{E}_{b}.$

Observe that under the conditions of Theorem 4.2,

$\displaystyle s_{\bm{a},k}^{{\sf
PCA}}\leq\frac{\sqrt{r}(\lambda_{\max}+\sigma^{2})}{\Delta_{\min}\sqrt{n}}+\frac{(\lambda_{j}+\sigma^{2})^{1/2}\sigma}{\lambda_{j}\sqrt{n}}\lesssim\frac{1}{\log(n\vee
p)},$

and hence it holds that
$|\bm{a}^{\top}\bm{\widehat{u}}_{k}|\lesssim|\bm{a}^{\top}\bm{u}_{k}|.$
Therefore, with probability at least $1-O((n\vee p)^{-8})$

$\displaystyle\bigg{|}(\bm{a}^{\top}\bm{\widehat{u}}_{k})^{2}(1+\widehat{b_{k}^{{\sf
PCA}}})-(\bm{a}^{\top}\bm{u}_{k})^{2}\bigg{|}$
$\displaystyle\lesssim|\bm{a}^{\top}\bm{u}_{k}|\bigg{(}s_{\bm{a},j}^{{\sf
PCA}}\frac{s_{\bm{a},k}^{{\sf PCA}}}{s_{\bm{a},j}^{{\sf PCA}}}\log(n\vee
p)+|\bm{a}^{\top}\bm{u}_{k}|\mathcal{E}_{b}\bigg{)}$

which completes the proof. ∎

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