# Large-Scale Cover Song Detection in Digital Music Libraries Using Metadata,
Lyrics and Audio Features

###### Abstract

Cover song detection is a very relevant task in Music Information Retrieval
(MIR) studies and has been mainly addressed using audio-based systems. Despite
its potential impact in industrial contexts, low performances and lack of
scalability have prevented such systems from being adopted in practice for
large applications. In this work, we investigate whether textual music
information (such as metadata and lyrics) can be used along with audio for
large-scale cover identification problem in a wide digital music library. We
benchmark this problem using standard text and state of the art audio
similarity measures. Our studies shows that these methods can significantly
increase the accuracy and scalability of cover detection systems on Million
Song Dataset (MSD) and Second Hand Song (SHS) datasets. By only leveraging
standard tf-idf based text similarity measures on song titles and lyrics, we
achieved 35.5% of absolute increase in mean average precision compared to the
current scalable audio content-based state of the art methods on MSD. These
experimental results suggests that new methodologies can be encouraged among
researchers to leverage and identify more sophisticated NLP-based techniques
to improve current cover song identification systems in digital music
libraries with metadata.

##  1 Introduction

From the advent of digital music platforms, music listening has been shifting
from physical storage devices to digital music streaming platforms. Thousands
of new songs are ingested daily to these platforms. Some music streaming
services claims to have more than 40 million of songs in their catalogs.
Hence, structuring and organizing songs in such large databases is of great
interest since it can help to improve music recommendation and user-experience
(to name a few). Beside that, understanding the underlying relations of a song
and its cover versions are also beneficial for musicological studies. Cover
song detection is a well identified task in MIR studies. It is typically set
as a query-retrieval task where the system retrieves a ranked list of possible
covers of a given query song from a reference database. Cover songs may have
variations in key, tempo, structure, melody, harmony, timbre, language and
lyrics compared to its original version, which makes it a challenging task.
Over the past decade, numerous studies have been done on this problem using
audio-based methods. An extensive review of traditional cover song detection
system can be found in [27]. Most of the best performing cover detection
systems according to MIREX111http://www.music-ir.org/mirex/wiki/ framework
rely on exposing the underlying tonal content of cover/non-cover pairs to
compute the music similarities within them since most of the cover songs
usually share considerable tonal similarity between each other.

Tonal features are mid-level audio features that correspond to the musical
tonality of a sound such as key, melody, harmony or chords. Among them, chroma
audio features have been widely used for the task of cover identification
since it is a timbre invariant and octave-independent representation of the
note-based tonal progression of a song over time. Variants of chroma features
such as Harmonic Pitch Class Profile (HPCP) [25, 28] and Chroma Energy
Normalized Statistics (CENS) [16] were commonly used by existing systems since
it gives an added robustness to tuning deviation, noise and local tempo
variations compared to a normal chroma feature. Beside chroma features, melody
(pitch salience) [23], chord profiles [12], self-similarity MFCC matrix [32],
cognition-inspired descriptors [34] were also used in the literature. A cover
song similarity distance is then computed using various sequence alignment
techniques such as dynamic time warping [25], smith-waterman algorithm [32],
recurrence quantification analysis [28], cross-correlation plots [8], SiMPle
[30] and information-theoretic measures [9]. A preprocessing step is usually
done to these tonal features before computing this distance in order to make
them invariant to the key or the tempo of the song. Key invariance can be
obtained by Optimal Transposition Index (OTI) [26] or by computing 2D-Fourier
transform magnitude coefficients of tonal feature vectors by disregarding the
phase coefficients [1]. Tempo invariance can be achieved by using beat-
synchronous chroma features[8, 32].

On the other hand, machine learning techniques have been less explored in
solving this task. In 2009, Ravuri et. al [22] used a SVM classifier fed with
different distance measures computed from chroma features at various tempos to
classify cover/non-cover pairs (best second performing system in MIREX). This
idea of combining similarity measures from different features was also further
explored in the recent works of [20, 32, 7]. In [10], the authors explored the
use of metric learning by projecting audio features in a high dimensional
space where simple distance measures capture cover song similarity. Recently
in [6], the authors trained a CNN with cross similarity matrices obtained from
CENS features of cover/non-cover pairs to predict the probability of whether a
reference song is a cover or non-cover of the query song. The idea of
efficient database pruning and thus achieving multi-stage architecture for
cover identification was explored in the works of [19] and [4]. Results
reported in MIREX evaluations show little to no improvements for cover
detection since 2009. The system proposed in [28, 25] still remains as the
best performing algorithms in MIREX with a mean average precision (MAP) of
$0.75$ on the MIREX mixed music collections. However MIREX cover detection
evaluation is done on a dataset of 1000 songs with pairwise comparisons
systems that are not scalable. To address this issue, several researchers
studied and formulated the problem of large-scale cover identification. But
the accuracy of these systems were not satisfactory (the best MAP reported is
$0.28$ on the Second Hand Song dataset (SHS) train set alone [11]). A more
detail review of scalable cover detection systems can be found in Section 2.
Recently there has been a growing interest in applying domain-specific
knowledge such as from NLP on solving traditional MIR tasks using multi-modal
approaches such as in genre classification [17] and music recommendation [18].
Even though NLP techniques were previously used in MIR studies for different
tasks such as lyrics-alignment [14], mood, tag, genre classification [13],
particularly for the cover detection problem, the reliability and the accuracy
of text-based approaches have not been studied or benchmarked beside in the
recent work of [31]. In this work, our main purpose is to benchmark the
problem of large-scale cover identification using multi-modal data such as
song title, lyrics and audio in a general framework considering all these
different modalities of data are available to us. We believe that the results
presented in this work could be a new direction for future industrial-scale
cover identification systems dealing with commercial digital music libraries
where textual, content and semantic informations of music are mostly
available. The remainder of this paper is organized as follows: we describe
the problem of large-scale cover identification, the related methodologies,
datasets and evaluation frameworks in Section 2. In Section 3, we explain the
setup of our experimental system. In Section 4, we discuss and evaluate the
proposed system in comparison to the state-of-the-art. Finally, we draw some
conclusions in Section 5.

##  2 Large-scale cover identification

###  2.1 Related works

Acknowledging the limits in terms of scalability and tractability of
traditional cover detection systems, Bertin-Mahieux et. al [2] used audio
fingerprinting-inspired hashing to compute jump-codes from beat-synchronous
chroma [8] to detect covers. Audio fingerprinting [5] methods were
successfully used in the industry to find an almost identical copies of a
given piece of music with high robustness to pitch, noise, distortions and
other transformations. Later, audio fingerprinting strategies were explored in
the works of [21, 33] for detecting live version of cover songs. Authors of
[1] and [11] tried to achieve the factor of scalability and invariance by
efficiently capturing some of this invariance in a low-dimensional feature
space using sparse representations where simple distance metrics give a notion
of cover song similarity. [11] reported highest MAP score on the large-scale
cover identification problem. Note that this is different to the non-scalable
MIREX based evaluations we discussed earlier. Another way to achieve
scalability in cover detection systems is by doing efficient database-pruning
using various features in a preliminary step. This enabled us to use audio
algorithms with more specificity to capture more understandable cover
similarity in the later layers. This idea of multi-stage architecture were
explored in the works of [31, 4, 20, 19] using audio based methods.

###  2.2 Datasets

As we talk about the large-scale cover identification problem, most of the
public datasets like
covers80222https://labrosa.ee.columbia.edu/projects/coversongs/covers80/ and
covers1000333http://www.covers1000.net/ are inadequate since it has only a few
hundred or thousands of songs. Identifying this problem, authors of [1] also
proposed the SHS dataset, a subset of MSD [3] which has almost eighteen
thousand songs clustered as cover cliques. Since it is a subset of MSD, the
entire MSD can be used to evaluate cover identification systems and thus make
it comparable to commercial scale applications. So far from our knowledge, SHS
and MSD are the only publicly available dataset for large-scale cover
identification tasks. The Musixmatch
(MXM)444https://labrosa.ee.columbia.edu/millionsong/musixmatch dataset
provides bag-of-words (BOW) lyrics for 23.76% songs in the MSD. Table 1 shows
the summary of datasets used throughout our experiments. Among them, the
mapping of MSD and Deezer audio database (MSD-DZR) were internally available
from Deezer555http://www.deezer.com which enabled us to run the audio-based
experiments. We further explain various existing and observed problems in
these datasets.

Dataset |  Description
---|---
MSD |  Metadata and precomputed audio features for 1 million songs
SHS |  18,196 MSD songs organized as cover cliques (available as train & test splits)
Musixmatch (MXM) |  Mapping of bag-of-words lyrics provided by Musixmatch for 237,662 songs of MSD
MSD-DZR |  Mapping of 82.45% MSD songs with the Deezer audio database
SHS-DZR |  Subset of MSD-DZR with corresponding mapping to SHS (with train and test splits)
Table 1: Summary of datasets used in this study.

####  2.2.1 Similarities within cover set

We did a preliminary analysis on understanding how the MSD song titles are
similar within songs in the same clique and other cliques. This concept of
understanding cover similarity within a cover clique were previously explored
through audio in [29]. Figure 1 shows the distribution of similarity measures
for all MSD song titles within and outside their respective cliques in the SHS
train set. This similarity measure is computed by the Levenshtein distance
ratio666https://github.com/ztane/python-Levenshtein between the two song
titles. Here 1 indicates maximum similarity and 0 indicates none. From the
Figure 1, we can clearly see a strong separability and discriminatory property
of song-title in differentiating covers in the case of SHS train set as
expected. This can be true for most of the popular music cover versions
without language invariance when metadata are properly attributed. For
example, the song “All Along The Watch Tower” by Bob Dylan has cover songs
with titles “All Along The Watchtower (Jimi Hendrix cover)”, “ Along The
Watchtower (2001 Digital Remaster)” etc. But this similarity in titles are not
preserved in lot of non-pop music. For example in classical music, a song in a
cover clique is often distinguished using unique identifiers like opus numbers
while most of the other strings in the title exhibits strong string similarity
with songs in other cover cliques (eg. Chopin Mazurka in C major Op. 7, No. 5
and Chopin Mazurka in C major Op. 24, No. 2 are two different works of art).
From our observation, different non-pop music like western classical music are
not well represented in the MSD. This is one of the limitations of evaluating
generic cover detection systems on SHS and MSD.

Figure 1: String similarity of MSD song titles in the SHS train set within and
outside its clique

####  2.2.2 Language invariance

One of the main challenges of using song titles and lyrics for cover retrieval
is the language variance. To understand the scope of this problem, we detect
the language of songs in MSD using their available lyrics or song title (in
case of no lyrics). From this process, almost 70 percent of songs in the MSD
were detected as in English, thanks to the open-source library language-
detection777https://github.com/shuyo/language-detection. A unknown is
returned, when no language is detected. From empirical observation we noticed
that among the 16% of unknown flagged songs, a considerable part of them are
actually in English. Beside that, approximately 3% of songs are in Spanish, 2%
in French and majority of the remaining in various European languages.
Conceptually, a simple way to achieve language invariance would be to
translate text into a common language, like English. While this approach
should be explored in future studies, we decided to ignore this problem due to
the low impact it will have on our mostly-English dataset. Note that this
English predominance may be a another drawback of SHS and MSD for evaluating
generic text-based cover detection systems.

####  2.2.3 Effects of duplicates and unannotated covers

The presence of duplicates and unannotated SHS covers within MSD are a known
problem within the MIR
community888https://labrosa.ee.columbia.edu/millionsong/secondhand. MSD
creators provide an official list of MSD
duplicates999https://labrosa.ee.columbia.edu/millionsong/blog/11-3-15-921810-song-
dataset-duplicates. Particularly in the case of cover song detection, this has
been explained in the work of [20]. This can create two problems while
evaluating cover identification systems with SHS and MSD datasets. The
presence of duplicates might give overly optimistic results when using audio-
based methods since query and reference tracks are sometimes identical and
thus yielding a low distance. To address the duplicate problem, we flagged the
official MSD duplicate tracks in our database to exclude them in our
experiments. The effect of official MSD duplicates in the final accuracy of
experiments are further discussed in Section 4. On the other hand, the
presence of unannotated covers can affect the measured performance of a cover
detection system even though it detects the actual covers of the query songs.
We found several examples of these cases while exploring the SHS train set
where there are many unannotated covers with same or very similar metadata and
lyrics as of a query song. This means that the evaluations of cover detection
systems on SHS and MSD might not exactly reflect the actual accuracy of the
proposed system.

##  3 Experimental setup

###  3.1 Methodology

In this Section, we detail our proposed experimental system. We introduce a
cover detection system which leverages song metadata, lyrics and audio
comprehensively. Regarding evaluation of our system, we followed the same
standard approach as in [1, 11], thus making it comparable to existing state
of the art large-scale cover detection systems. We report MAP [15] at top-k
results as evaluation metric for our experiments since it reflects the
accuracy of covers identified in high ranks. Other metrics such as mean
reciprocal rank, mean rank of first cover and average rank are not found to be
reliable since we use top-k results of the search response. We did our
exploration experiments on the SHS train set (12,960 tracks) and kept the SHS
test set (5,238 tracks) aside for the final evaluation as in [11].

ID | Description
---|---
title | search by MSD-song-title
pre-title | search by pre-processed MSD-song-title
mxm-lyr | search by MXM-lyrics
+ | denotes merging and re-ranking process
Table 2: Summary of different methods used in our experiments

###  3.2 Text-based approaches

In this section, we further explain the methods used to retrieve text-based
cover song similarity from song titles and lyrics. Here, similarity of text is
obtained by computing the cosine similarity of weighted vectors, where the
weighted vectors are the term frequency(tf) and inverted document
frequency(idf) computed from the document. term frequency is a count of how
many times a word appear in a given bag-of-words document and inverse document
frequency is the number of times a word occurs in a corpus of documents. We
used the standard search and text engine Apache
Lucene101010https://lucene.apache.org/ for computing this text similarity.
Lucene uses a combination of Boolean Space (BM) and Vector Space Model (VSM)
of information retrieval [24] for computing this cosine similarity. Readers
are referred to lucene scoring
formula111111https://lucene.apache.org/core/4_9_0/core/org/apache/lucene/search-/similarities/TFIDFSimilarity.html
for further details on the performance optimization functions adapted in
Lucene. tf-idf-based search is a standard process in information retrieval and
well adapted in all modern search engines. Moreover, tf-idf methods can be
easily adapted to the bag-of-words lyrics provided in the MXM dataset.

####  3.2.1 Metadata-based approach

From the analysis in Section 2.2.1, we already see that the song titles are a
potential discriminatory feature to detect cover songs in the case of SHS
train set. Cover songs titles usually have certain keywords like live, cover,
ep, studio, acoustic, remix, remastered, version etc to often describe various
types of cover. From our exploration on SHS train set we noticed that the
presence of these keywords often introduce a variance in the song titles among
a cover clique. Hence, we stored a preprocessed MSD title as a new metadata
field, removing this invariance. We define a codebook of these common keywords
(total 34 keywords) by exploring the SHS train set (eg. version, remaster,
live, edit, album, demo, reprise, edit, radio, original, cover, single etc).
We parsed strings appearing inside parenthesis in MSD song titles. The parsed
string is then removed from the song title if a possible match has been found
with its stemmed version and any keywords in the defined codebook. This
preprocessing step of keywords is very similar to the flagging approach done
in [31]. 11.38 % of songs in the whole MSD and 12.42 % of songs in the SHS
train set were affected by this preprocessing step. Our experiments on the SHS
train sets achieved better accuracy on pre-processed MSD titles than the
original ones, as one can read from Table 3. This shows that removing non-
relevant information from song titles might help to increase the accuracy of
retrieving covers. For example, searching by the song title ”Be (1994 Digital
Remastered Version)” using tf-idf-based methods leads to noisy results found
from the words 1994, digital, remastered and version.

Metadata in real world music databases can be untidy and prone to errors. In
these cases, customized text processing strategies can be employed. The artist
credits associated with a work of art (e.g. Composer, Lyricist…etc) can also
be a relevant metadata for cover detection in some cases. However, this
valuable metadata is less likely to be found in a large corpus. For instance,
MSD does not contain such metadata. This is something to be investigated in
future works.

####  3.2.2 Lyrics-based approach

In the case of cover songs excluding instrumental versions, lyrics encode
certain semantic features about a common work of art. Especially in the case
of pop-song covers, the lyric content tends to be the same or very similar
between a song and its cover. Lyrics retrieval have been previously studied by
means of various NLP based alignment techniques[14]. For our experiments, we
decided to use a bag-of-words (BOW) tf-idf-based retrieval as we explained in
the beginning of this section. By this experiment we aim to check whether
lyrics helps in some way to retrieve it’s cover songs. We used the more like
this121212https://www.elastic.co/guide/en/elasticsearch/reference/current/query-
dsl-mlt-query.html query method in Elasticsearch
(ES)131313https://www.elastic.co/products/elasticsearch to retrieve songs with
similar lyrics to avoid reinventing the wheel with custom methods since we
want to have a baseline result. This query method selects the top-k terms with
highest tf-idf to form a disjunctive query of these terms to retrieve similar
documents. We finally used the parameters max_query_terms=12 and
min_query_terms=1 of this method by experimenting with various examples in the
SHS train set. On the other hand, the lyrics provided by the MXM dataset are
also distributed as BOW format which makes it ideal for us to use MXM lyrics
in the experiments.

MAP@k | k=10 | k=100
---|---|---
title | 0.346 | 0.397
pre-title | 0.35 | 0.401
mxm-lyr | 0.137 | 0.140
title + mxm-lyr | 0.376 | 0.433
pre-title + mxm-lyr | 0.372 | 0.431
Table 3: Results of experiments with SHS train set against whole MSD.

###  3.3 Data framework

Figure 2: Data framework used throughout our experiments with textual data

We use ES as database to store and serve all textual metadata and lyrics used
in our large-scale experiments. ES is built on top of the Lucene search engine
which makes it an ideal candidate for standard information retrieval and text
processing functions. Each track from the MSD was ingested in the ES index,
selecting only the metadata used in our experiments. Data from other datasets
such as SHS and MXM were then added to existing document, in an update
operation. The matching between these datasets and MSD are publicly provided
by their respective creators. The overall data framework can be visualized in
Figure 2. During ingestion of textual data, the ES standard analyzer divides
text into terms on word boundaries by Unicode Text
Segmentation141414http://unicode.org/reports/tr29/ algorithm (this is similar
to a stemming process). Readers are referred to ES query-dsl
documentation151515https://www.elastic.co/guide/en/elasticsearch/reference/current/query-
dsl.html on further understanding the various search functions available
within ES.

###  3.4 Audio-based approaches

The preliminary results in Table 3 and Table 4 on SHS train set and MSD
reveals that metadata-based pruning achieves higher results than the scalable
audio-based approaches in the previous works. We decided to use the
traditional audio-based cover detection system proposed in [29] with a aim to
further improve the accuracy achieved by text-based methods: this algorithm
achieved the best MAP score (0.75) for this MIREX campaign till date and the
executable files are publicly
available161616http://joanserra.weebly.com/publications.html. The output of
this algorithm is a distance matrix from pairwise similarities of all the
query-reference song pairs in the dataset. A small distance means the
reference track is more likely to be a cover of the queried song. A multi-
modal framework as proposed in our work allows the evaluation of such a strong
audio-based algorithm in large-scale. This means that instead of computing
pairwise music similarities of 1 against 999,999 tracks of MSD, we can process
pairwise distances of a query against pruned top-k results from the text-based
methods. We did an analysis to estimate this optimum pruning size for text-
based approaches from where we can further apply more specific audio-based
methods. As we can see in Figure 3, the MAP metric does not evolve
significantly after the top-100 pruning. From this observation, we can safely
rely on the top-100 pruned results to further perform audio-based cover
detection [29] and thus having a trade-off between accuracy and scalability.
Effects of this proposed methods are further detailed in Section 4.

Figure 3: MAP@k plot for MSD song title experiment with SHS train set against
the whole MSD.

###  3.5 Combining meta-data, lyrics and audio results

In this section we explain our methods to comprehensively combine the results
from song title, lyrics and audio similarity measures. This can be seen as
similar to the multi-layer (stage) cover detection frameworks as discussed in
[31] and [4]. MAP score increases when actual covers are detected in the top
ranks. Thus using content-based features such as lyrics to re-rank the
response from song title experiments should increase the MAP of the system.
Table 2 shows the summary of different methods we used in the experiments.
Here, we obtained the title + mxm-lyr merged list by concatenating common
documents present in the title and mxm-lyr results on the top ranks and the
rest of documents in the title response in the low ranks. Thus we obtain a new
merged list title + mxm-lyr. By doing so, we minimize some possible errors of
the merging process without penalizing the MAP achieved with the title method.
Note that for some optimization, we only consider the documents with high
scores in the mxm-lyr results for the above mentioned merging process. These
documents are selected by applying a relative threshold h on the difference of
its score at rank 1 against the score at all other ranks in the mxm-lyr
results. A default threshold parameter h=0.5 was used in our experiments after
exploring with various values on the lyrics responses. Table 3 shows the
results of text-based experiments on SHS train set against the whole MSD. We
can see that re-ranking title with mxm-lyr results boosts the accuracy of
title method by 3.6% in the case of top 100 results of SHS train set against
MSD experiment. Here, we have to take in account that only 23.76% of
corresponding lyrics to MSD were available in the MXM dataset. So in this
merging process, the system still maintains the accuracy obtained from the
title method even though there is no lyrical data available for the queried
song. This is quite similar to the real world scenarios since music lyrics can
be comparatively hard to find in non-commercial music databases. Regarding
audio methods, our chosen system [29] returns a very low distance value (which
is close to zero) when the query and reference pairs have high tonal music
similarity. Here we obtain the new merged list by re-ranking the best
performing text-based method (title + mxm-lyr) results by only using the songs
with low music similarity distances (as given by [29]) to the top ranks. We
used absolute thresholding to select these songs with low distance scores. A
default absolute threshold h=$0.1$ was used in our final experiments which was
found empirically by exploring the training data. We can see whole this
process as a three stage architecture which can be optimally designed using
more novel methods in the future works.

| MAP
---|---
Osmalskyj et. al[20] | 0.048
Bertin-Mahieux et. al [1] | 0.09
Khadkevich et. al[12] | 0.10
Humphrey et. al[11] | 0.28
title | 0.829
pre-title | 0.927
mxm-lyr | 0.139
title + mxm-lyr | 0.921
pre-title + mxm-lyr | 0.926
Table 4: Results of experiments with SHS train set against itself.

##  4 Final evaluation

We now display the results corresponding to the final evaluation of our
methods on the unseen data (SHS test set) against the whole MSD. We used the
same evaluation methodologies as in [1] and [11]. Table 5 compares the MAP
scores of all our proposed text-based methods against the previous state of
the art works. Note that we report MAP@100 scores of our experiments with and
without the official MSD duplicates and MAP@100 is always less than MAP. From
Table 5, we can see a 32.5% absolute increase in accuracy by only using MSD
song title compared to the current audio-based state of the art [11].This is
very interesting considering the fact that song title information is widely
and easily available in public compared to other data such as lyrics. Among
all, title + mxm-lyr method achieved the best MAP score of 0.489 (35.5%
absolute increase to [11]). Another interesting thing to notice is that the
MSD song title preprocessing step explained in Section 3 does not necessarily
improve the results on the SHS test set. This shows that more sophisticated
preprocessing strategies or text topic-modeling approaches can be borrowed in
future works.

Having discussed about the impact of various text-based methods on the
experiments with MSD, SHS train and test sets, our next aim is to observe how
far the state of the art cover detection algorithms can further improve the
accuracy of text-based methods. Table 6 shows the results of various
experiments done on the SHS-DZR test set (4,254 tracks) against MSD-DZR
(824,585 tracks) set by excluding all the official MSD duplicates. Note that
the baseline results in Table 6 are computed on the official SHS and MSD
datasets, which is slightly different. Confirming this observation and
quantifying the results we can see that re-ranking text-based (title + mxm-
lyr) results using state of the art audio-method [29] with low distance
thresholds (h=0.1) further boosts the MAP score by 2.1%.

| MAP | MAP (no dupl.)
---|---|---
Bertin-Mahieux et al [1] | 0.029 | –
Humphrey et. al[11] | 0.134 | –
title | 0.421 | 0.459
pre-title | 0.421 | 0.453
mxm-lyr | 0.148 | 0.149
title + mxm-lyr | 0.442 | 0.489
pre-title + mxm-lyr | 0.442 | 0.486
Table 5: Results of experiments with SHS test set against whole MSD. | MAP
---|---
Bertin-mahieux et al [1] | 0.029
Humphrey et. al[11] | 0.134
title | 0.478
title + mxm-lyr | 0.510
pre-title + mxm-lyr | 0.506
title + mxm-lyr + Serra et. al [29] (h=0.1) | 0.531
Table 6: Results of experiments with audio re-ranking on SHS-DZR test set
against MSD-DZR

##  5 Conclusions

In this study we successfully demonstrated that combining song metadata,
lyrics and audio features can significantly improve accuracy and scalability
of cover song identification in a simple and easy-to-set-up architecture. We
report the best MAP score on the MSD to date with a 35.5% absolute increase
with title + mxm-lyr method and 32.5% absolute increase using only MSD song
titles compared to the current audio content-based state of the art methods.
To our knowledge, our work reports the first benchmark results on
metadata/lyrics-based cover song detection using SHS, MSD and MXM datasets. It
is quite important to notice that the results presented in this work addresses
the use-case of cover detection in a digital music library where corresponding
metadata, lyrics and audio are mostly available such as in the case of music
streaming services like Deezer. Hence, our method is not competing with
traditional audio-based cover identification methods in the literature when
none of the metadata or textual content like lyrics are not available.
However, song titles are a widely available music metadata and its high
performance as a unique input for cover song identification is indeed
promising for stimulating further scientific research.To an extent, similarity
of song titles and lyrics within cover sets are more interpretable than from
audio considering the musicological aspects involved in audio feature
computation. Involving text topic-modelling and machine learning models could
be explored as an immediate direction for future research on text-based cover
detection.

The main drawback of our method is the loss of some actual covers in the
pruned results when cover song pairs have entirely different metadata or
lyrical content. In these scenarios, audio-based methods can give better
results. On the other hand, considering the case of music metadata catalogs
where the audio content is not available, text-based cover song detection
methods can be highly beneficial in database structuring, musicological
studies etc. Hence, an ideal cover detection system should be able to leverage
the strength of multi-modal methods according to the use-case of its
applications and thus helping to identify strong generic audio-based cover
detection systems.

Our work is built on top of previous works using public datasets like MSD, SHS
and MXM thus making it comparable and easily reproducible (in terms of text-
based methods). Finally, for open research and reproducibility all our code
and data used for text-based methods are available on-line in a public
repository171717https://github.com/deezer/cover_song_detection.

##  6 Acknowledgements

The authors would like to thank Dr. Francesco Piccoli for the interesting
discussions and reviews during this work.

The authors would also like to thank everyone in the Deezer R&D team for their
valuable comments and helpful suggestions. This work was funded and supported
by Deezer S.A, Paris, France. The research leading to this work benefited from
the WASABI project supported by the French National Research Agency (contract
ANR-16-CE23-0017-01).

## References

  * [1] Thierry Bertin-Mahieux and Daniel P W Ellis.  Large-scale cover song recognition using the 2d fourier transform magnitude.  International Society for Music Information Retrieval, 2012.
  * [2] Thierry Bertin-Mahieux and Daniel P.W. Ellis.  Large-scale cover song recognition using hashed chroma landmarks.  IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, 2011.
  * [3] Thierry Bertin-Mahieux, Daniel P.W. Ellis, Brian Whitman, and Paul Lamere.  The million song dataset.  In Proceedings of the 12th International Conference on Music Information Retrieval (ISMIR 2011).
  * [4] Kang Cai, Deshun Yang, and Xiaoou Chen.  Two-layer large-scale cover song identification system based on music structure segmentation.  2016 IEEE 18th International Workshop on Multimedia Signal Processing, MMSP 2016, 2017.
  * [5] Pedro Cano, Eloi Batle, Ton Kalker, and Jaap Haitsma.  A review of algorithms for audio fingerprinting.  In Multimedia Signal Processing, 2002 IEEE Workshop on, pages 169–173. IEEE, 2002.
  * [6] Sungkyun Chang, Juheon Lee, Sang Keun Choe, and Kyogu Lee.  Audio cover song identification using convolutional neural network.  arXiv preprint arXiv:1712.00166, 2017.
  * [7] Ning Chen, Wei Li, and Haidong Xiao.  Fusing similarity functions for cover song identification.  Multimedia Tools and Applications, 77(2):2629–2652, 2018.
  * [8] Daniel PW Ellis and Graham E Poliner.  Identifyingcover songs’ with chroma features and dynamic programming beat tracking.  In Acoustics, Speech and Signal Processing, 2007. ICASSP 2007. IEEE International Conference on. IEEE, 2007.
  * [9] Peter Foster, Simon Dixon, and Anssi Klapuri.  Identifying cover songs using information-theoretic measures of similarity.  IEEE/ACM Transactions on Audio, Speech and Language Processing (TASLP), 23(6):993–1005, 2015.
  * [10] Hoon Heo, Hyunwoo J Kim, Wan Soo Kim, and Kyogu Lee.  Cover song identification with metric learning using distance as a feature.  2017\.
  * [11] Eric J. Humphrey, Oriol Nieto, and Juan P. Bello.  Data Driven and Discriminative Projections for Large-scale Cover Song Identification.  Proc. of the 14th International Society for Music Information Retrieval Conference, 2013.
  * [12] Maksim Khadkevich and Maurizio Omologo.  Large-Scale Cover Song Identification Using Chord Profiles.  Proceedings of the 14th International Society for Music Information Retrieval Conference (ISMIR-2013), 2013.
  * [13] Peter Knees and Markus Schedl.  Music Similarity and Retrieval: An Introduction to Audio-and Web-based Strategies, volume 36.  Springer, 2016.
  * [14] Peter Knees, Markus Schedl, and Gerhard Widmer.  Multiple lyrics alignment: Automatic retrieval of song lyrics.  In International Society for Music Information Retrieval Conference (ISMIR), 2005.
  * [15] Christopher D Manning, Prabhakar Raghavan, Hinrich Schütze, et al.  Introduction to information retrieval, volume 1.  Cambridge university press Cambridge, 2008.
  * [16] Meinard Müller, Frank Kurth, and Michael Clausen.  Audio matching via chroma-based statistical features.  In Proceedings of the 6th International Conference on Music Information Retrieval (ISMIR), 2005.
  * [17] Sergio Oramas, Oriol Nieto, Francesco Barbieri, and Xavier Serra.  Multi-label Music Genre Classification from Audio, Text, and Images Using Deep Features.  International Conference on Music Information Retrieval (ISMIR), 2017.
  * [18] Sergio Oramas, Oriol Nieto, Mohamed Sordo, and Xavier Serra.  A Deep Multimodal Approach for Cold-start Music Recommendation.  Proceedings of the 2nd Workshop on Deep Learning for Recommender Systems - DLRS 2017.
  * [19] J. Osmalskyj, S. Piérard, M. Van Droogenbroeck, and J. J. Embrechts.  Efficient database pruning for large-scale cover song recognition.  In 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pages 714–718, 2013.
  * [20] Julien Osmalskyj, Peter Foster, Simon Dixon, and Jean-jacques Embrechts.  Combining features for cover song identification.  16th International Society for Music Information Retrieval Conference (ISMIR), 2015.
  * [21] Z. Rafii, B. Coover, and J. Han.  An audio fingerprinting system for live version identification using image processing techniques.  In 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 644–648, 2014.
  * [22] Suman V. Ravuri and Daniel P. W. Ellis.  Cover song detection: From high scores to general classification.  In Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2010, Texas, USA, 2010.
  * [23] Justin Salamon, Joan Serra, and Emilia Gómez.  Tonal representations for music retrieval: from version identification to query-by-humming.  International Journal of Multimedia Information Retrieval, 2(1):45–58, 2013.
  * [24] Gerard Salton, Anita Wong, and Chung-Shu Yang.  A vector space model for automatic indexing.  Communications of the ACM, 18(11):613–620, 1975.
  * [25] J. Serra, E. Gomez, P. Herrera, and X. Serra.  Chroma binary similarity and local alignment applied to cover song identification.  IEEE Transactions on Audio, Speech, and Language Processing, 2008\.
  * [26] Joan Serrà, Emilia Gómez, and Perfecto Herrera.  Transposing chroma representations to a common key.  IEEE CS Conference on The Use of Symbols to Represent Music and Multimedia Objects, 2008.
  * [27] Joan Serrà, Emilia Gómez, and Perfecto Herrera.  Audio cover song identification and similarity: Background, approaches, evaluation, and beyond.  Studies in Computational Intelligence, 274, 2010.
  * [28] Joan Serra, Xavier Serra, and Ralph G. Andrzejak.  Cross recurrence quantification for cover song identification.  New Journal of Physics, 2009.
  * [29] Joan Serrà, Massimiliano Zanin, and R. G. Andrzejak.  Cover song retrieval by cross recurrence quantification and unsupervised set detection.  In Music Information Retrieval Evaluation eXchange (MIREX) extended abstract, 2009.
  * [30] Diego F Silva, Chin-Chin M Yeh, Gustavo Enrique de Almeida Prado Alves Batista, Eamonn Keogh, et al.  Simple: Assessing music similarity using subsequences joins.  In International Society for Music Information Retrieval Conference, 2016.
  * [31] J. B. L. Smith, M. Hamasaki, and M. Goto.  Classifying derivative works with search, text, audio and video features.  In 2017 IEEE International Conference on Multimedia and Expo (ICME), July 2017.
  * [32] Christopher J Tralie.  Early mfcc and hpcp fusion for robust cover song identification.  International Conference on Music Information Retrieval (ISMIR), 2017.
  * [33] T. Tsai, T. Prätzlich, and M. Müller.  Known-artist live song identification using audio hashprints.  IEEE Transactions on Multimedia, 2017.
  * [34] JMH van Balen, Dimitrios Bountouridis, Frans Wiering, and Remco C Veltkamp.  Cognition-inspired descriptors for scalable cover song retrieval.  In proceedings of the 15th international conference on Music Information Retrieval, 2014.


# Ricci Flow Equation on $(\alpha,\beta)$-Metrics

A. Tayebi, E. Peyghan and B. Najafi

###### Abstract

In this paper, we study the class of Finsler metrics, namely
$(\alpha,\beta)$-metrics, which satisfies the un-normal or normal Ricci flow
equation.

Keywords: Finsler metric, Einstein metric, Ricci flow equation.111 2010
Mathematics subject Classification: 53B40, 53C60.

##  1 Introduction

In 1982, R. S. Hamilton for a Riemannian metric $g_{ij}$ introduce the
following geometric evolution equation

$\frac{d}{dt}(g_{ij})=-2Ric_{ij},\ \ g(t=0)=g_{0},$

where $Ric_{ij}$ is the Ricci curvature tensor and is known as the un-
normalised Ricci flow in Riemannian geometry [7]. Hamilton showed that there
is a unique solution to this equation for an arbitrary smooth metric on a
closed manifold over a sufficiently short time. He also showed that Ricci flow
preserves positivity of the Ricci curvature tensor in three dimensions and the
curvature operator in all dimensions [6]. The Ricci flow theory related
geometric analysis and various applications became one of the most intensively
developing branch of modern mathematics [5, 7, 8, 12, 17]. The most important
achievement of this theory was the proof of W. Thurston’s geometrization
conjecture by G. Perelman [14, 15, 16]. The main results on Ricci flow
evolution were proved originally for (pseudo) Riemannian and Kähler
geometries. Thus the Ricci flow theory became a very powerful method in
understanding the geometry and topology of Riemannian and Kählerian manifolds.

On the other hand, Finsler geometry is a natural extension of Riemannian
geometry without quadratic restriction [19]. But it is not simple to define
Ricci flows of mutually compatible fundamental geometric structures on Finsler
manifolds. The problem of constructing the Finsler-Ricci flow theory contains
a number of new conceptual and fundamental issues on compatibility of
geometrical and physical objects and their optimal configurations. The same
equation can be used in the Finsler setting, since both the fundamental tensor
$g_{ij}$ and Ricci tensor $Ric_{ij}$ have been generalized to that broader
framework, albeit gaining a $y$ dependence in the process [2][18]. However,
there are some reasons why we shall refrain from doing so: (i) not every
symmetric covariant 2-tensor $g_{ij}(x,y)$ arises from a Finsler metric
$F(x,y)$; (ii) there is more than one geometrical context in which $g_{ij}$
makes sense. Thus, Bao called this equation as an un-normalised Ricci flow for
Finsler geometry. Using the elegance work of Akbar-Zadeh in [1], Bao proposed
the following normalised Ricci flow equation for Finsler metrics

$\frac{d}{dt}\log F=-R+\frac{1}{Vol(SM)}\int_{SM}R\ dV,\ \ \ F(t=0)=F_{0},$
(1)

where the underlying manifold $M$ is compact [2].

In a series of papers, Vacaru studied Ricci flow evolutions of geometries and
physical models of gravity with symmetric and nonsymmetric metrics and
geometric mechanics, when the field equations are subjected to nonholonomic
constraints and the evolution solutions, mutually transform as Riemann and
Finsler geometries [20, 21, 22, 23, 24, 25, 26].

It is remarkable that, Chern had asked whether every smooth manifold admits a
Ricci-constant Finsler metric? The weaker case of this question is that
whether every smooth manifold admits a Einstein Finsler metric? His question
has already been settled in the affirmative for dimension 2 because, by a
construction of Thurston s, every Riemannian metric on a two-dimensional
manifold admits a complete Riemannian metric of constant Gaussian curvature.

##  2 Preliminaries

Let $M$ be an $n$-dimensional $C^{\infty}$ manifold. Denote by $T_{x}M$ the
tangent space at $x\in M$, and by $TM=\cup_{x\in M}T_{x}M$ the tangent bundle
of $M$.

A Finsler metric on $M$ is a function $F:TM\rightarrow[0,\infty)$ which has
the following properties:
(i) $F$ is $C^{\infty}$ on $TM_{0}:=TM\setminus\\{0\\}$;
(ii) $F$ is positively 1-homogeneous on the fibers of tangent bundle $TM$,
(iii) for each $y\in T_{x}M_{0}$, the following form $g_{y}$ on $T_{x}M$ is
positive definite,

$g_{y}(u,v):={1\over 2}\left[F^{2}(y+su+tv)\right]|_{s,t=0},\ \ u,v\in
T_{x}M.$

For a Finsler metric $F=F(x,y)$ on a manifold $M$, the spray ${\bf
G}=y^{i}\frac{\partial}{\partial x^{i}}-2G^{i}\frac{\partial}{\partial y^{i}}$
is a vector field on $TM$, where $G^{i}=G^{i}(x,y)$ are defined by

$G^{i}={g^{il}\over
4}\Big{\\{}[F^{2}]_{x^{k}y^{l}}y^{k}-[F^{2}]_{x^{l}}\Big{\\}},$

Let $x\in M$ and $F_{x}:=F|_{T_{x}M}$. To measure the non-Euclidean feature of
$F_{x}$, define ${\bf C}_{y}:T_{x}M\otimes T_{x}M\otimes
T_{x}M\rightarrow\mathbb{R}$ by

${\bf C}_{y}(u,v,w):={1\over 2}\frac{d}{dt}\left[{\bf
g}_{y+tw}(u,v)\right]|_{t=0},\ \ u,v,w\in T_{x}M,$

The family ${\bf C}:=\\{{\bf C}_{y}\\}_{y\in TM_{0}}$ is called the Cartan
torsion. It is well known that ${\bf{C}}=0$ if and only if $F$ is Riemannian.
For $y\in T_{x}M_{0}$, define mean Cartan torsion ${\bf I}_{y}$ by ${\bf
I}_{y}(u):=I_{i}(y)u^{i}$, where $I_{i}:=g^{jk}C_{ijk}$,
$C_{ijk}=\frac{1}{2}\frac{{\partial}g_{ij}}{{\partial}y^{k}}$ and
$u=u^{i}\frac{\partial}{\partial x^{i}}|_{x}$. By Deicke’s Theorem, $F$ is
Riemannian if and only if ${\bf I}_{y}=0$.

Regarding the Cartan tensors of these metrics, M. Matsumoto introduced the
notion of C-reducibility and proved that any Randers metric $F=\alpha+\beta$
and Kropina metric $F=\alpha^{2}/\beta$ are C-reducible, where
$\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric and
$\beta=b_{i}(x)y^{i}$ is a 1-form on $M$. Matsumoto-Hōjō proved that the
converse is true [10]. Furthermore, by considering Kropina and Randers
metrics, Matsumoto introduced the notion of $(\alpha,\beta)$-metrics [9]. An
$(\alpha,\beta)$-metric is a Finsler metric on $M$ defined by
$F:=\alpha\phi(s)$, where $s=\beta/\alpha$, $\phi=\phi(s)$ is a $C^{\infty}$
function on the $(-b_{0},b_{0})$ with certain regularity, $\alpha$ is a
Riemannian metric and $\beta$ is a 1-form on $M$.

In [11], Matsumoto-Shibata introduced the notion of semi-C-reducibility by
considering the form of Cartan torsion of a non-Riemannian
$(\alpha,\beta)$-metric on a manifold $M$ with dimension $n\geq 3$. A Finsler
metric is called semi-C-reducible if its Cartan tensor is given by

$C_{ijk}={\frac{p}{1+n}}\\{h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}\\}+\frac{q}{C^{2}}I_{i}I_{j}I_{k},$

where $p=p(x,y)$ and $q=q(x,y)$ are scalar function on $TM$,
$h_{ij}:=g_{ij}-F^{-2}y_{i}y_{j}$ is the angular metric and
$C^{2}=I^{i}I_{i}$. If $q=0$, then $F$ is just C-reducible metric.

##  3 Ricci Flow Equation

In 1982, R. S. Hamilton introduce the following geometric evolution equation

$\frac{d}{dt}(g_{ij})=-2Ric_{ij},\ \ g(t=0)=g_{0}$

which is known as the un-normalised Ricci flow in Riemannian geometry [7]. The
same equation can be used in the Finsler setting, since both the fundamental
tensor $g_{ij}$ and Ricci tensor $Ric_{ij}$ have been generalized to that
broader framework, albeit gaining a $y$ dependence in the process. However,
there are some reasons why we shall refrain from doing so: (i) Not every
symmetric covariant 2-tensor $g_{ij}(x,y)$ arises from a Finsler metric
$F(x,y)$; (ii) There is more than one geometrical context in which $g_{ij}$
makes sense. Thus, Bao called this equation as an un-normalised Ricci flow for
Finsler geometry.

Professor Chern had asked, on several occasions, whether every smooth manifold
admits a Ricci-constant Finsler metric. It is hoped that the Ricci flow in
Finsler geometry eventually proves to be viable for addressing Chern s
question. How to formulate and generalize these constructions for non-
Riemannian manifolds and physical theories is a challenging topic in
mathematics and physics. Bao studied Ricci flow equation in Finsler spaces
[2]. In the following a scalar Ricci flow equation is introduced according to
the Bao’s paper.

A deformation of Finsler metrics means a 1-parameter family of metrics
$g_{ij}(x,y,t)$, such that $t\in[-\epsilon,\epsilon]$ and $\epsilon>0$ is
sufficiently small. For such a metric $\omega=u_{i}dx^{i}$, the volume element
as well as the connections attached to it depend on $t$. The same equation can
be used in the Finsler setting. We can also use another Ricci flow equation
instead of this tensor evolution equation [2]. By contracting
$\frac{d}{dt}g_{ij}=-2Ric_{ij}$ with $y^{i}$ and $y^{j}$ gives, via Euler s
theorem, we get

$\frac{\partial F^{2}}{\partial t}=-2F^{2}R,$

where $R=\frac{1}{F^{2}}Ric$. That is,

$d\log F=-R,\ \ F(t=0)=F_{0}.$

This scalar equation directly addresses the evolution of the Finsler metric
$F$, and makes geometrical sense on both the manifold of nonzero tangent
vectors $TM_{0}$ and the manifold of rays. It is therefore suitable as an un-
normalized Ricci flow for Finsler geometry.

##  4 Un-Normal Ricci Flow Equation on $(\alpha,\beta)$-Metrics

Here, we study $(\alpha,\beta)$-metrics satisfying un-normal Ricci flow
equation and prove the following.

######  Theorem 4.1.

Let $(M,F)$ be a Finsler manifold of dimension $n\geq 3$. Suppose that
$F=\Phi(\frac{\beta}{\alpha})\alpha$ be an $(\alpha,\beta)$-metric on $M$.
Then every deformation $F_{t}$ of the metric $F$ satisfying un-normal Ricci
flow equation is an Einstein metric.

To prove the Theorem 4.1, we need the following.

######  Lemma 4.2.

Let $F_{t}$ be a deformation of an $(\alpha,\beta)$-metric $F$ on manifold $M$
of dimension $n\geq 3$. Then the variation of Cartan tensor is given by
following

$\displaystyle
C^{\prime}_{ijk}I^{i}I^{j}I^{k}=-\frac{2R(1+nq)}{1+n}||I||^{4}-\frac{1}{2}F^{2}R_{,i,j,k}I^{i}I^{j}I^{k}-3||I||^{2}I^{m}R_{,m}$
(2)

where $||I||^{2}=I_{m}I^{m}$.

###### Proof.

First assume that $F_{t}$ be a deformation of a Finsler metric on a two-
dimensional manifold $M$ satisfies Ricci flow equation, i.e.

$\frac{d}{dt}g_{ij}:=g^{\prime}_{ij}=-2Ric_{ij},\ \ \ d\log
F:=\frac{F^{\prime}}{F}=-R,$ (3)

where $R=\frac{1}{F^{2}}Ric$. By definition of Ricci tensor, we have

$\displaystyle Ric_{ij}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ \frac{1}{2}[RF^{2}]_{y^{i}y^{j}}$ (4)
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
Rg_{ij}+\frac{1}{2}F^{2}R_{,i,j}+R_{,i}y_{j}+R_{,i}y_{i}$

where $R_{,i}=\frac{\partial R}{\partial y^{i}}$ and
$R_{,i,j}=\frac{\partial^{2}R}{\partial y^{i}\partial y^{j}}$. Taking a
vertical derivative of (4) and using $y_{i,j}=g_{ij}$ and $FF_{k}=y_{k}$
yields

$\displaystyle Ric_{ij,k}=2RC_{ijk}+\frac{1}{2}F^{2}R_{,i,j,k}\\!\\!\\!\\!$
$\displaystyle+$ $\displaystyle\\!\\!\\!\\!\
\\{g_{jk}R_{,i}+g_{ij}R_{,k}+g_{ki}R_{,j}\\}$ (5) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\ \\{R_{,j,k}y_{i}+R_{,i,j}y_{k}+R_{,k,i}y_{j}\\}.$

Contracting (5) with $I^{i}I^{j}I^{k}$ and using $y_{i}I^{i}=y^{i}I_{i}=0$
implies that

$\displaystyle
Ric_{ij,k}I^{i}I^{j}I^{k}=2RC_{ijk}I^{i}I^{j}I^{k}+\frac{1}{2}F^{2}R_{,i,j,k}I^{i}I^{j}I^{k}+3||I||^{2}I^{m}R_{,m}.$
(6)

The Cartan tensor of an $(\alpha,\beta)$-metric on a $n$-dimensional manifold
$M$ is given by

$C_{ijk}={\frac{p}{1+n}}\\{h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}\\}+\frac{q}{||I||^{2}}I_{i}I_{j}I_{k},$
(7)

where $p=p(x,y)$ and $q=q(x,y)$ are scalar function on $TM$ with $p+q=1$.
Multiplying (7) with $I^{i}I^{j}I^{k}$ yields

$C_{ijk}I^{i}I^{j}I^{k}=({\frac{p}{1+n}}+q)||I||^{4}=\frac{1+nq}{1+n}||I||^{4}.$
(8)

Then by (6) and (8), we get

$\displaystyle
Ric_{ij,k}I^{i}I^{j}I^{k}=\frac{2R(1+nq)}{1+n}||I||^{4}+\frac{1}{2}F^{2}R_{,i,j,k}I^{i}I^{j}I^{k}+3||I||^{2}I^{m}R_{,m}.$
(9)

On the other hand, since $F_{t}$ satisfies Ricci flow equation then

$\displaystyle C^{\prime}_{ijk}=\frac{1}{2}\frac{\partial
g^{\prime}_{ij}}{\partial
y^{k}}=\frac{1}{2}\frac{\partial(-2Ric_{ij})}{\partial y^{k}}=-Ric_{ij,k}.$
(10)

By (9) and (10) we get (2). ∎

######  Lemma 4.3.

Let $F_{t}$ be a deformation of an $(\alpha,\beta)$-metric $F$ on a
$n$-dimensional manifold $M$. Then $C^{\prime}_{ijk}I^{i}I^{j}I^{k}$ is a
factor of $||I||^{2}$.

###### Proof.

Since $g^{ij}g_{jk}=\delta^{i}_{k}$, then we have

$0=(g^{ij}g_{jk})^{\prime}=g^{\prime ij}g_{jk}+g^{ij}g^{\prime}_{jk}=g^{\prime
ij}g_{jk}-2g^{ij}Ric_{jk},$

or equivalently $g^{\prime ij}g_{jk}=2g^{ij}Ric_{jk}$ which contracting it
with $g^{lk}$ implies that

$g^{\prime il}=2Ric^{il}.$ (11)

Then we have

$\displaystyle I^{\prime}_{i}=(g^{jk}C_{ijk})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
(g^{jk})^{\prime}C_{ijk}+g^{jk}C^{\prime}_{ijk}$ (12) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2Ric^{jk}C_{ijk}-g^{jk}Ric_{jk,i}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
2Ric^{jk}g_{jk,i}-(g^{jk}Ric_{jk})_{,i}+g^{jk}_{\ \ ,i}Ric_{jk}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\ -(g^{jk}Ric_{jk})_{,i}=-\rho_{i}$

where $\rho:=g^{jk}Ric_{jk}$ and $\rho_{i}=\frac{\partial\rho}{\partial
y^{i}}$. Thus

$\displaystyle I^{\prime i}=(g^{ij}I_{j})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
(g^{ij})^{\prime}I_{j}+g^{ij}I^{\prime}_{j}$ (13) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2Ric^{ij}I_{j}-g^{ij}\rho_{j}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2Ric^{ij}I_{j}-\rho^{i}.$

The variation of $y_{i}:=FF_{y^{i}}$ with respect to $t$ is given by

$y^{\prime}_{i}=-2Ric_{im}y^{m}.$

Therefore, we can compute the variation of angular metric $h_{ij}$ as follows

$\displaystyle h^{\prime}_{ij}=(g_{ij}-F^{-2}y_{i}y_{j})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-2Ric_{ij}-2F^{-2}Ry_{i}y_{j}+2F^{-2}(Ric_{im}+Ric_{jm})y^{m}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-2Ric_{ij}+2R(h_{ij}-g_{ij})+2(Ric_{im}\ell_{j}+Ric_{jm}\ell_{i})\ell^{m},$

where $\ell_{i}:=F^{-1}y_{i}$. Thus

$h^{\prime}_{ij}=2Rh_{ij}-2Rg_{ij}-2Ric_{ij}+2(Ric_{im}\ell_{j}+Ric_{jm}\ell_{i})\ell^{m}.$
(14)

Now, we consider the variation of Cartan tensor

$\displaystyle C^{\prime}_{ijk}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\
\big{\\{}\frac{p}{1+n}[h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}]+\frac{q}{||I||^{2}}I_{i}I_{j}I_{k}\big{\\}}^{\prime}$
(15) $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
\frac{p^{\prime}}{1+n}[h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}]+\frac{q^{\prime}}{||I||^{2}}I_{i}I_{j}I_{k}$
$\displaystyle+$ $\displaystyle\\!\\!\\!\\!\
\frac{p}{1+n}\Big{[}h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}\Big{]}^{\prime}+q\Big{[}\frac{1}{||I||^{2}}I_{i}I_{j}I_{k}\Big{]}^{\prime}$

We have

$[\frac{1}{||I||^{2}}I_{i}I_{j}I_{k}]^{\prime}=\frac{-(I^{\prime
m}I_{m}+I^{m}I^{\prime}_{m})}{||I||^{2}}C_{ijk}-\frac{1}{||I||^{2}}(\rho_{i}I_{j}I_{k}+\rho_{j}I_{i}I_{k}+\rho_{k}I_{i}I_{j})$
(16)

Multiplying (16) with $I^{i}I^{j}I^{k}$ implies that

$\displaystyle[\frac{1}{||I||^{2}}I_{i}I_{j}I_{k}]^{\prime}I^{i}I^{j}I^{k}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\ -\big{[}\frac{(nq+1)(I^{\prime
m}I_{m}+I^{m}I^{\prime}_{m})}{(n+1)||I||^{2}}+3\rho_{m}I^{m}\big{]}||I||^{2}$
(17) $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
\big{[}\frac{2(nq+1)(\rho^{m}I_{m}-Ric^{pq}I_{p}I_{q})}{(n+1)||I||^{2}}-3\rho_{m}I^{m}\big{]}||I||^{2}$

On the other hand

$\displaystyle[h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}]^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-(\rho_{i}h_{jk}+\rho_{j}h_{ik}+\rho_{k}h_{ij})-2R[I_{i}g_{jk}+I_{j}g_{ik}+I_{k}g_{ij}]$
(18) $\displaystyle-$ $\displaystyle\\!\\!\\!\\!\
2[I_{i}Ric_{jk}+I_{j}Ric_{ik}+I_{k}Ric_{ij}]$ $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\ 2R[I_{i}h_{jk}+I_{j}h_{ik}+I_{k}h_{ij}]$
$\displaystyle+$ $\displaystyle\\!\\!\\!\\!\
2[I_{i}\Lambda_{jk}+I_{j}\Lambda_{ik}+I_{k}\Lambda_{ij}],$

where $\Lambda_{jk}:=(Ric_{jr}\ell_{k}+Ric_{kr}\ell_{j})\ell^{r}$. Multiplying
(18) with $I^{i}I^{j}I^{k}$ implies that

$\displaystyle[h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j}]^{\prime}I^{i}I^{j}I^{k}=-3(\rho_{m}I^{m}+2Ric_{pq}I^{p}I^{q})||I||^{2}.$
(19)

On the other hand, since $p^{\prime}+q^{\prime}=0$ then we get

$\displaystyle\Big{[}\frac{p^{\prime}}{1+n}(h_{ij}I_{k}+h_{jk}I_{i}+h_{ki}J_{j})+\frac{q^{\prime}}{||I||^{2}}I_{i}I_{j}I_{k}\Big{]}I^{i}I^{j}I^{k}=\frac{nq^{\prime}}{1+n}||I||^{4}$
(20)

Putting (17), (19) and (20) in (15) implies that
$C^{\prime}_{ijk}I^{i}I^{j}I^{k}$ is a factor of $||I||^{2}$. More precisely,
we have the following

$\displaystyle
C^{\prime}_{ijk}I_{i}I_{j}I_{k}=\Big{[}\frac{nq^{\prime}}{n+1}||I||^{2}\\!\\!\\!\\!$
$\displaystyle-$ $\displaystyle\\!\\!\\!\\!\
q\Big{(}\frac{2(nq+1)(\rho^{m}I_{m}-Ric^{pq}I_{p}I_{q})}{(n+1)||I||^{2}}-3\rho_{m}I^{m}\Big{)}$
(21) $\displaystyle-$ $\displaystyle\\!\\!\\!\\!\
\frac{3p}{n+1}(\rho_{m}I^{m}+2Ric_{pq}I^{p}I^{q})\Big{]}||I||^{2}.$

This completes the proof. ∎

Proof of Theorem 4.1: By Lemmas 4.2 and 4.3, it follows that
$R_{,i,j,k}I^{i}I^{j}I^{k}$ is a factor of $||I||^{2}$. Thus

$R_{,i,j,k}I^{i}I^{j}I^{k}=A_{ij}I_{k}+B_{i}g_{jk}.$

It is remarkable that, since $R_{,i,j,k}$ is symmetric with respect to indexes
i, j and k, then the order of indexes in this relation doesn t matter. Now,
multiplying $R_{,i,j,k}$ with $y^{k}$ or $y^{j}$ implies that $R_{,i}=0$. It
means that $R=R(x)$ and then $F_{t}$ is an Einstein metric.

##  5 Normal Ricci Flow Equation on $(\alpha,\beta)$-Metrics

If M is a compact manifold, then $S(M)$ is compact and we can normalize the
Ricci flow equation by requiring that the flow keeps the volume of $SM$
constant. Recalling the Hilbert form $\omega:=F_{y^{i}}dx^{i}$, that volume is

$Vol_{SM}:=\int_{SM}\frac{(-1)}{(n-1)!}^{\frac{n(n-1)}{2}}\omega\wedge(d\omega)^{n-1}:=\int_{SM}dV_{SM}.$

During the evolution, $F$, $\omega$ and consequently the volume form $dV_{SM}$
and the volume $Vol_{SM}$, all depend on $t$. On the other hand, the domain of
integration $SM$, being the quotient space of $TM_{0}$ under the equivalence
relation $z\sim y$ , $z=\lambda y$ for some $\lambda>0$, is totally
independent of any Finsler metric, and hence does not depend on t. We have

$\frac{d}{dt}(dV_{SM})=\big{[}g_{ij}\frac{d}{dt}g_{ij}-n\frac{d}{dt}\log
F\big{]}dV_{SM}$

A normalized Ricci flow for Finsler metrics is proposed by Bao as follows

$\frac{d}{dt}\log F=-R+\frac{1}{Vol(SM)}\int_{SM}R\ dV,\ \ \ F(t=0)=F_{0},$
(22)

where the underlying manifold $M$ is compact. Now, we let $Vol(SM)=1$. Then
all of Ricci-constant metrics are exactly the fixed points of the above flow.
Let

$Ric_{ij}=\frac{1}{2}(F^{2}R)_{.y^{i}.y^{j}}$

and differentiating (22) with respect to $y^{i}$ and $y^{j}$ the following
normal Ricci flow tensor evaluation equation is concluded

$\frac{d}{dt}g_{ij}=-2Ric_{ij}+\frac{2}{Vol(SM)}\int_{SM}R\ dVg_{ij},\ \ \
g(t=0)=g_{0},$ (23)

Starting with any familiar metric on M as the initial data $F_{0}$, we may
deform it using the proposed normalized Ricci flow, in the hope of arriving at
a Ricci constant metric.

######  Theorem 5.1.

Let $(M,F)$ be a Finsler manifold of dimension $n\geq 3$. Suppose that
$F=\Phi(\frac{\beta}{\alpha})\alpha$ be an $(\alpha,\beta)$-metric on $M$.
Then every deformation $F_{t}$ of the metric $F$ satisfying normal Ricci flow
equation is an Einstein metric.

###### Proof.

Now, we consider Finsler surfaces that satisfies the normal Ricci flow
equation. Then

$\frac{dg_{ij}}{dt}=-2Ric_{ij}+2\int_{SM}R\ dVg_{ij},\ \ d\log
F:=\frac{F^{\prime}}{F}=-R+\int_{SM}R\ dV.$ (24)

By the same argument used in the un-normal Ricci flow case, we can calculate
the variation of mean Cartan tensor as follows

$\displaystyle I^{\prime}_{i}=(g^{jk}C_{ijk})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
(g^{jk})^{\prime}C_{ijk}+g^{jk}C^{\prime}_{ijk}$ (25) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2[Ric^{jk}-\int_{SM}R\
dVg^{jk}]C_{ijk}+g^{jk}[Ric_{jk,i}+2\int_{SM}R\ dVC_{ijk}]$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2Ric^{jk}g_{jk,i}-(g^{jk}Ric_{jk})_{,i}+g^{jk}_{\
\ ,i}Ric_{jk}$ $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-(g^{jk}Ric_{jk})_{,i}$ $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
-\rho_{i}$

Then we have

$\displaystyle I^{\prime i}=(g^{ij}I_{j})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\
(g^{ij})^{\prime}I_{j}+g^{ij}I^{\prime}_{j}$ (26) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\ 2[Ric^{ij}-\int_{SM}R\
dVg^{ij}]I_{j}-g^{ij}\rho_{j}$

By the same way that we used in un-normal Ricci flow, it follows that

$\displaystyle C^{\prime}_{ijk}=\frac{-(I^{\prime
m}I_{m}+I^{m}I^{\prime}_{m})}{||I||^{2}}C_{ijk}-\frac{1}{||I||^{2}}(\rho_{i}I_{j}I_{k}+\rho_{j}I_{i}I_{k}+\rho_{k}I_{i}I_{j})$
(27)

Contracting it with $I^{i}I^{j}I^{k}$ yields

$C^{\prime}_{ijk}I^{i}I^{j}I^{k}=(\Omega||I||^{2}-3\rho_{m}I^{m})||I||^{2},$
(28)

where

$\Omega:=-\frac{I^{\prime
m}I_{m}+I^{m}I^{\prime}_{m}}{||I||^{2}}=\frac{2\rho^{m}I_{m}-2Ric^{ml}I_{l}}{||I||^{2}}+2\int_{SM}R\
dV.$

By Lemma 4.2, we deduce that $R_{,i,j,k}I^{i}I^{j}I^{k}$ is a factor of
$||I||^{2}$. By the same argument, it result that every deformation $F_{t}$ of
the metric $F$ satisfying normal Ricci flow equation is an Einstein metric. ∎

## References

  * [1] H. Akbar-Zadeh, Generalized Einstein manifolds, J. Geom. Phys. 17(1995), 342-380.
  * [2] D. Bao, On two curvature-driven problems in Riemann-Finsler geometry, Advanced Studies in Pure Mathematics XX, 2007.
  * [3] D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry. In A Sampler of Riemann Finsler Geometry, Mathematical Sciences Research Institute Publications, vol. 50, Cambridge University Press, 2004, 197-259.
  * [4] L. Berwald, On Cartan and Finsler Geometries, III, Two Dimensional Finsler Spaces with Rectilinear Extremal, Ann. Math. 42 No. 2 (1941) 84 122.
  * [5] H. -D. Cao and X. -P. Zhu, Hamilton–Perelman’s proof of the Poincar’e conjecture and the geometrization conjecture, Asian J. Math. 10 (2006) 165–495; arXiv: math. DG/ 0612069.
  * [6] R. S. Hamilton, Four Manifolds with Positive Curvature Operator, J. Diff. Geom. 24, 153-179, 1986.
  * [7] R. S. Hamilton, Three–manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255-306.
  * [8] B. Kleiner and J. Lott, Notes on Perelman’s papers, arXiv: math.DG/ 0605667.
  * [9] M. Matsumoto, Theory of Finsler spaces with $(\alpha,\beta)$-metric, Rep. Math. Phys. 31(1992), 43-84.
  * [10] M. Matsumoto and S. Hōjō, A conclusive theorem for C-reducible Finsler spaces, Tensor. N. S. 32(1978), 225-230.
  * [11] M. Matsumoto and C. Shibata, On semi-C-reducibility, T-tensor and S4-1ikeness of Finsler spaces, J. Math. Kyoto Univ. 19(1979), 301-314.
  * [12] J. W. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, arXiv: math.DG/ 0607607.
  * [13] B. Najafi and A. Tayebi, On a family of Einstein Randers metric, International Journal of Geometric Methods in Modern Physics, (2011), DOI No: 10.1142/S021988781100552X.
  * [14] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/ 0211159.
  * [15] G. Perelman, Ricci flow with surgery on three–manifolds, arXiv: math. DG/ 03109.
  * [16] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three–manifolds, arXiv: math.DG/ 0307245.
  * [17] N. SadeghZadeh and A. Razavi, Ricci Flow equation on C-reducible metrics, International Journal of Geometric Methods in Modern Physics, (2011), DOI No: 10.1142/S0219887811005385
  * [18] A. Tayebi and E. Peyghan, On Ricci tensors of Randers metrics, J. Geom. Phys. 60(2010), 1665-1670.
  * [19] A. Tayebi and E. Peyghan, On a class of Riemannian metrics arising from Finsler structures, C. R. Acad. Sci. Paris, Ser. I 349 (2011), 319-322.
  * [20] S. Vacaru, Nonholonomic ricci flows: II. evolution equations and dynamics, J. Math. Phys.  49 (2008), 043504.
  * [21] S. Vacaru, Nonholonomic Ricci flows, exact solutions in gravity, and Symmetric and Nonsymmetric Metrics, Int. J. Theor. Phys.  48  (2009) 579-606
  * [22] S. Vacaru, The Entropy of Lagrange-Finsler spaces and Ricci flows, Rep. Math. Phys.  63  (2009) 95-110
  * [23] S. Vacaru, Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows, J. Math. Phys.  50  (2009) 073503
  * [24] S. Vacaru, Fractional Nonholonomic Ricci Flows, arXiv:1004.0625.
  * [25] S. Vacaru, Nonholonomic Ricci flows and parametric deformations of the solitonic pp–waves and Schwarzschild solutions, Electronic Journal of Theoretical Physics 6, N21 (2009), 63-93.
  * [26] S. Vacaru, Nonholonomic Ricci flows: Exact solutions and gravity, Electronic Journal of Theoretical Physics 6, N20 (2009) 27-58.

Akbar Tayebi
Faculty of Science, Department of Mathematics
Qom University
Qom. Iran
Email: akbar.tayebi@gmail

Esmail Peyghan
Faculty of Science, Department of Mathematics
Arak University
Arak. Iran
Email<EMAIL_ADDRESS>
Behzad Najafi
Faculty of Science, Department of Mathematics
Shahed University
Tehran. Iran
Email<EMAIL_ADDRESS>