Forensic Analysis of Video Files Using Metadata

Ziyue Xiang^⋆   János Horváth^⋆   Sriram Baireddy^⋆
Paolo Bestagini^†   Stefano Tubaro^†   Edward J. Delp^⋆

^⋆ Video and Image Processing Lab (VIPER), School of Electrical and Computer Engineering,

Purdue University, West Lafayette, Indiana, USA

^† Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy

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©2021 IEEE.  Personal use of this material is permitted.  Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

The unprecedented ease and ability to manipulate video content has led to a rapid spread of manipulated media.
The availability of video editing tools greatly increased in recent years, allowing one to easily generate photo-realistic alterations.
Such manipulations can leave traces in the metadata embedded in video files.
This metadata information can be used to determine video manipulations, brand of video recording device, the type of video editing tool, and other important evidence.
In this paper, we focus on the metadata contained in the popular MP4 video wrapper/container.
We describe our method for metadata extractor that uses the MP4's tree structure.

Our approach for analyzing the video metadata produces a more compact representation.

We will describe how we construct features from the metadata and then use
dimensionality reduction and nearest neighbor classification for forensic analysis of a video file.
Our approach allows one to visually inspect the distribution of metadata features and make decisions.
The experimental results confirm that the performance of our approach surpasses other methods.

§ INTRODUCTION

The proliferation of easy-to-use video manipulation tools has placed unprecedented power in the hands of individuals.
Recently, an Indian politician used deepfake technology to rally more voters [22].
In the original video the politician delivered his message in English; it was convincingly altered to show him speaking in local dialect.
Media manipulation methods are also used as tools of criticism and to undermine the reputation of politicians [12].

Such manipulated videos can now be easily generated to bolster disinformation campaigns and sway the public opinion on critical issues.

A wide variety of tools for video forensic analysis have been developed [26].
These tools can be used to attribute a video to the originating device, to reconstruct the past video compression history, and even to detect video manipulations.
The most popular video manipulation detection techniques focus on inconsistencies and artifacts in the pixel domain [6, 24, 27, 29].
As video manipulation detection methods become more sophisticated, video editing techniques continue to improve, leading to a situation where manipulated videos are becoming practically indistinguishable from real videos [7, 8, 14, 28, 36, 38].
For this reason, detection techniques exploiting pixel-level analysis may fail, while methods that do not use pixel data will increasingly gain importance.

Video forensic techniques not exploiting pixel analysis typically work due to the presence of “metadata” [15, 20].
This is additional embedded information that every video file contains.
The metadata are used for video decoding [21] and indicating other information such as the date, time, and location of the video when created.
Because video editing tools tend to cause large structural changes in metadata, it is difficult for one to alter a video file without leaving any metadata traces.

Therefore, metadata can serve as strong evidence in video forensics tasks.

In this paper, we leverage the seminal work presented in [20, 39] to investigate the use of metadata for video forensic analysis of the MP4 and MOV video formats, which are among the most popular video wrappers/containers.

The MP4 format is used by numerous Android devices, social networks, and digital video cameras [13, 33, 41].

MOV format is mostly used by Apple devices and is derived from the same ISO standard as MP4 [18].

The design of the MP4 format is based on MOV [2].
The two formats can be parsed in a similar manner, thus we will refer to MP4 containers hereinafter even if MOV containers are considered.
As a result, our approach can analyze a large number of videos in the real world.

In our work, we examine the results of using the metadata in MP4 files for video forensic scenarios, extending the work presented in [39].
More specifically, we describe a metadata extraction method and improve the feature representation format to make metadata-based forensic feature vectors more compact.
We employed feature selection techniques to boost the quality of the feature vectors.
Finally, we reduced the dimensionality of the feature vectors to two, which allows visualization and classification in 2D space.

We show that these feature vectors can be used for a wide variety of video forensic tasks, from source attribution to tampering detection.
Compared to other work, our proposed approach can generate 2D feature scatter plots and decision boundary graphs for many video forensics tasks.
This feature enables us to gain insights into the distribution of MP4 metadata and make interpretable decisions.

Our experimental results show that many video forensics problems on standard datasets can be solved reliably by looking only at metadata.
We also discovered that videos uploaded to specific social networks (e.g., TikTok, WeiBo) present altered metadata, which invalidates metadata-based forensics methods.
This is one of the limitations of our techniques and will be addressed in future research.

§ RELATED WORK

Many techniques have been described to determine whether some part of a video has been tampered or not [26].
Most of these methods were developed to detect manipulations in the pixel domain and do not use metadata information.
Compared to pixel-level analysis, metadata-based methods possess unique advantages.
The size of metadata is significantly smaller than pixel data, which enables the analysis of large datasets in short amounts of time.
Most video manipulation software do not allow users to alter metadata directly [15, 20].
Consequently, metadata has a higher degree of opacity than pixel data, which makes metadata-based media forensics more reliable and its corresponding attacks more challenging.

Most existing work focuses on the metadata in MP4-like video containers, which maintain data in a tree structure.
In [20] and [39], the authors design features based on symbolic representation of MP4's tree structure, which are processed by a statistical decision framework and decision trees, respectively.
The authors report high performance for both video authentication and attribution tasks.
Güera  [15] extract video metadata with the |ffprobe| utility and then do video authentication with an ensemble classifier.

More low-level metadata-related information can be found by looking into video compression traces.
Video compression  methods typically leave series of traces related to the way the video frames are compressed.
This information is not easy to modify, thus acting as a metadata-like feature.
As an example, most contemporary video encoders compress frame sequences in a structures known as a Group of Pictures (GOP), where one frame can be defined using contents of other frames in order to save space [30].

The dependency between frames within or across different GOPs may provide evidence for video manipulation.
Due to the complexity of video codecs, a number of techniques have been proposed for various settings of a codec where specific video encoding features are turned on or off.
Vázquez-Padín  [37] provide a detailed explanation of the video encoding process and GOP structure.
They propose a video authentication method that generalizes across multiple video encoding settings.
Yao  [40] discuss the detection of double compression when an advanced video encoding feature called adaptive GOP is enabled.

§ PROPOSED APPROACH

§.§ An Overview of Our Approach

Video metadata captures multiple aspects of the history of a video file.
In this paper we propose a framework that exploits an MP4 video file's metadata to solve multiple video forensics problems, including brand attribution, video editing tool identification, social network attribution, and manipulation detection.
Our method can also be easily adapted to other forensics applications.

The structure of our proposed framework is illustrated in Figure <ref>. As will be discussed in Section <ref>, the MP4 format manages data using a tree structure. First we extract the metadata from MP4 files while preserving their relationships in the tree structure. The MP4 standard is around twenty years old, it contains numerous vendor-specific nuances that require separate parsing strategies. The metadata tree needs to go through several refining stages, which increase the granularity of the extracted information. In the next step, the tree representation of metadata is converted into a numeric feature vector, which can be easily processed by machine learning methods. Our feature representation scheme is based upon [39].

We improve the handling of media tracks and metadata fields that take on continuous values inside the tree.

The resulting feature vectors preserve more characteristics of the videos, yet they tend to also be more compact.

In the last stage, we use these features with a classifier based on the selected forensic application.
In the following we provide additional details about each step of our proposed framework.

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[stepnd] (s1) Metadata tree parsing;
[stepnd, right=of s1] (s2) |udta|
[stepnd, right=of s2] (s3) |meta|
[stepnd, right=of s3] (s4) |uuid|
[fit=(s1)(s2)(s3)(s4), fill=blue!10, inner sep=3mm, outer sep=0mm] (b1) ;
[anchor=north east] at (b1.north east) Comprehensive Video Metadata Extraction;
at (b1.center)  1;

[stepnd, below left=of b1, text width=4cm, anchor=west, xshift=15mm, yshift=-3mm] (s5) Vector representation of metadata tree;
[stepnd, right=of s5, text width=4cm, minimum height=2em] (s6) Feature selection;

(s5top) at ($(s5.north)+(0mm,5.5mm)$);
[-latex] (s4) |- (s5top) – (s5.north);

[fit=(s5), fill=blue!10, inner sep=3mm, outer sep=0mm] (b2) ;
at (b2.center)  2;

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at (b3.center)  3;

[stepnd, below=of s5, anchor=west, text width=6cm, yshift=-3mm, xshift=-3mm] (s7) Dimensionality reduction and classification;
(s7top) at ($(s7.north)+(0mm,4.5mm)$);
[-latex] (s6) |- (s7top) – (s7.north);

[fit=(s7), fill=blue!10, inner sep=3mm, outer sep=0mm, minimum width=] (b4) ;
at (b4.center)  4;

[node distance=1cm and 2mm]
[stepnd, below left=of b4, yshift=-3mm, xshift=7mm, text width=2cm, anchor=west] (s8) Brand attribution;
[stepnd, right=of s8, text width=2cm] (s9) Manipulation tool identification;
[stepnd, right=of s9, text width=2cm] (s10) Social network attribution;
[stepnd, right=of s10, text width=2cm] (s11) Manipulation detection;

[fit=(s8)(s9)(s10)(s11), fill=blue!10, inner sep=3mm, outer sep=0mm, minimum width=] (b4) ;
at (b4.center)  5;

The structure of our proposed metadata forensic analysis technique.

§.§ Video Metadata

The first step in our approach consists of parsing metadata from video files.
Digital video files are designed to handle multimodal data such as video, audio, and subtitles.
The standards of these data modalities also shifts as technology advances.
For example, since MP4's introduction in 2001, the mainstream video codec has changed from MPEG-2 to H.264, and may change to H.265 in the near future [17, 34, 35].

The metadata surrounding these various data modalities and standards are inserted into a video file in distinct ways. In this paper, we use the word comprehensive to describe a video metadata extraction scheme that is capable of parsing metadata across most data modalities and encoding specifications.

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[cellnd, draw=none, anchor=north west] at (_loc:nn 27) …;
(mp4t) at (_loc:nn 01);
(mp4b) at (_loc:nn 51);

(moovt) at (_loc:nn 111);
(moovb) at (_loc:nn 411);

[decorate,decoration=brace,amplitude=10pt,mirror, transform canvas=xshift=-1mm]
(mp4t) – (mp4b) node [black,midway,anchor=east,xshift=-4mm] MP4 file;
[decorate,decoration=brace,amplitude=10pt, transform canvas=xshift=1mm]
(moovt) – (moovb) node [black,midway,anchor=west,xshift=4mm] |moov| box;

Illustration of the MP4 file format, where each cell represents one byte. An MP4 file is made up of a series of boxes. Every box has an 8-byte header, where the first 4 bytes store the size of the box in bytes (including the header) as a 32-bit big-endian integer, and the last 4 bytes store the name of the box. The content of a box is either child boxes or binary data. Binary data from different boxes may require distinct decoding methods.

Figure <ref> shows the basic structure of an MP4 file [19].

An MP4 file is composed by a number of boxes. Each box starts with a header that specifies the box size (in byte) and name.
The size helps the user to locate box boundaries, and the name defines the purpose of the box. The content of a box can either be a child box or some encoded binary data.
Thus, an MP4 file is effectively a tree structure, where information is stored on leaf nodes.

Given MP4's tree structure, we can capture metadata information at two levels:

* the structure of the tree; and

* the interpreted values of binary data contained in leaf nodes
Therefore, the job of a metadata extractor is to traverse the MP4 tree, attain the tree's structure, filter non-metadata leaf nodes (e.g., nodes that contain video compressed pixel information) and interpret values on relevant leaves.
As shown in Figure <ref>, the output of a metadata extractor can be represented without any loss by a collection of human-readable strings.

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title=STRING ∫_use:N _string_box_int,
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at (segw) M;
[anchor=north west,
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The |moov| box is present in MP4 file.

The |mvhd| box is present in MP4 file. It is a child box of |moov|. One can also imply that the |moov| node itself must not contain any binary data.

The |mvhd| box is a leaf node in the tree; it contains a field called |duration|, whose value is 1546737.

|moov/udta/@$\bybox{A9}$mod=$\bybox{00}\bybox{09}$   $\bybox{81}$iPhone 5c|
The |udta| box is a leaf node in the tree; it contains a field called |$\bybox{A9}$mod|, whose value is |$\bybox{00}\bybox{09}$   $\bybox{81}$iPhone 5c|.

Examples of representing MP4 metadata tree with strings. Node paths are separated with `|/|', the values of leaf nodes are prefixed with `|@|', non-ASCII and unprintable characters are shown as hexadecimal codes surrounded by black frames. The metadata tree of any MP4 file can be portrayed by a collection of such strings.

Our metadata extractor focuses on vendor-specific non-standard MP4 metadata boxes that are significant for forensic purposes.
More precisely, we determine that |udta|, |uuid|, |meta|, and |ilst| boxes are likely to carry vital information for video forensics.
We next discuss our strategies to refine the parsing process.

§.§.§ Parsing |ilst| Data

|ilst| (“metadata item list”) boxes in MP4 files are used to store vendor-specific metadata [3].
Generally speaking, |ilst| boxes are container boxes whose child boxes carry metadata items as key-value pairs.
The names of |ilst|'s children (i.e., the keys) would start with |$\bybox{A9}$| (equivalent to character `©').
A list of frequently used |ilst| keys is shown in Table <ref>.
One can see that the content of the |ilst| box is particularly important for forensic analysis, for it often contains information about the manufacturer of the video capturing device, the encoder version, and the location and time of the capture.

A list of common keys in |ilst| boxes [4].

Key     Description
|$\bybox{A9}$mod|     camera model

|$\bybox{A9}$too|     name and version of encoding tool

|$\bybox{A9}$nam|     title of the content

|$\bybox{A9}$swr|     name and version number of creation software

|$\bybox{A9}$swf|     name of creator

|$\bybox{A9}$day|     timestamp of the video

|$\bybox{A9}$xyz|     geographic location of the video

It is difficult to parse the |ilst| box because various device manufacturers employ vastly different approaches when using it.
Below, we report some interesting variants we found during our experiments:

* |ilst|'s child boxes directly placed in |moov/udta|

In some old Apple devices (e.g., iPhone 4, iPhone 5c, iPad 2), the child boxes of |ilst| are placed directly in |moov/udta| box.

* |ilst| boxes in |moov/meta|

As its name suggests, the |meta| box is used to store metadata.
In this case, the |meta| box behaves similarly as other standard boxes, which means it can be parsed normally.
As the MP4 parser traverses the box, it will eventually reach |ilst| and its children.

* |ilst| boxes in |moov/udta/meta| and |moov/trak/meta|

When |meta| boxes appear in |udta| and |trak| boxes, they deviate from standard boxes.
More specifically, 4 extra bytes are inserted right after the |meta| header to store information [25].
These types of |meta| boxes cannot be parsed by the MP4 parser, normally because the program will see these 4 bytes as the size of next box, which will lead to corrupted results.

Our comprehensive metadata extractor is able to distinguish between these three scenarios and process MP4 video files correctly by fine-tuning the parsing of |udta| and |meta| boxes.

§.§.§ Parsing XML Data

We concluded that many video files contain XML data after inspecting numerous video files, especially those edited by ExifTool and Adobe Premiere Pro.
These tools make use of the Extensible Metadata Platform (XMP) to enhance metadata management, which introduces a large amount of metadata inside an MP4 file's |uuid| and |udta| boxes in the form of XML.
In Figure <ref>, we show two XML examples extracted from MP4 containers. It can be seen that these XML data can potentially contain a large amount of information, which includes type of manipulation, original value before manipulation, and even traces to locate the person that applied the manipulation.
It is vital for our metadata extractor to have the ability to handle XML data inside MP4 files.

To avoid over-complicating the extracted metadata tree, we discard the tree structure of XML elements and flatten them into a collection of key-value pairs.
If there is a collision between key names, the older value will be overwritten by the newer one, which indicates that only the last occurring value of each key is preserved.
Despite the fact that some information is lost by doing so, the data we have extracted is likely to be more than enough for most automated video forensic applications.

listing only,
listing engine=minted,
minted language=HTML,
minted options=

[Excerpt of XML data from a video processed by Adobe Premiere Pro. It clearly contains multiple important timestamps, the software name and version, and even the path to the Premiere project.]
xmlisting, listing file=listing/premiere.xml
[XML data from a video modified by ExifTool. It can be seen that the version of ExifTool is 11.37; the presence of |exif:DateTimeOriginal| implies the date of the video is modified.]
xmlisting, listing file=listing/exiftool.xml

Examples of XML data in MP4 video containers.

§.§ Track and Type Aware Feature

The second step of our approach consists of turning the parsed metadata into feature vectors.
Most machine learning methods use vectors as the input. The string representation of metadata trees generated in the previous step needs to be transformed into feature vectors before being used by machine learning methods.

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[node distance=3mm]

[metand] (m1) at (0, 1) |moov/mvhd|;
[metand, below=of m1.west, anchor=west] (m2) |moov/mvhd|;
[metand, below=of m2.west, anchor=west] |moov/mvhd|;

[fit=(m1)(m2), outer sep=0mm, inner sep=0mm] (m_12) ;
[above left=of m_12, anchor=west, xshift=2mm] number of occurrences of nodes;

[-latex] (m_12)-| (vec5) node [midway, above] $\featind{i}$;

[metand, anchor=west, xshift=-8mm] (m3) at (0, -1) |moov/trak/tkhd/@track_ID=1|;
[metand, below=of m3.west, anchor=west] (m4) |moov/trak/tkhd/@track_ID=1|;
[fit=(m3)(m4), outer sep=0mm, inner sep=0mm] (m_34) ;
[below left=of m_34, anchor=west, text width=4cm, yshift=-3mm, xshift=2mm] number of occurrences of categorical metadata fields;

[-latex] (m_34)– (vec3.south) node [midway, left] $\featind[2][d]{j}$;

[metand, anchor=west] (m6) at (4, -1<EMAIL_ADDRESS>[below left=of m6, anchor=west, text width=5cm, yshift=-3mm, xshift=2mm] value of continuous metadata fields;
[-latex] (m6)– (vec7.south) node [midway, right, xshift=2mm] $\featind[2][c^{\prime}]{k}$;

Illustration of the vector representation of MP4 metadata. The raise=1pt,scale=1.01.1$\chi$ functions help determine the corresponding element of a node or a metadata field in the feature vector.

Our feature representation technique is shown in Figure <ref>. For feature vectors to contain sufficient information of the MP4 tree, they need to include two levels of details: structure of the tree and value of metadata fields.

Metadata can be either categorical or continuous numerical fields. Considering categorical values, we assign each node and metadata key-value pair in the MP4 tree an element in the feature vector, which counts the number of occurrences of that node or pair. This strategy preserves information about the MP4 tree in the feature vector to a great extent.

Considering numerical values, creating a new element for each of these key-value pairs will render the feature vectors large and redundant. We decide to insert the values into the feature vectors directly.

From Figure <ref>, we know that string representation can be put into two categories:

* strings that indicate the presence of a node, with node path separated by `|/|'; and

* strings that show the key-value pair stored in a node, with node path separated by `|/|' and key-value separated by `|=|'.

Since an MP4 file can be seen as a collection of such strings, the feature transformation process can be viewed as a mapping from a given collection of strings $S$ to a vector $\bm{v}$.
For the discussion below, we use $\vecele{v}{l}$ to denote the $l$-th element of $\bm{v}$.

In order to construct a mapping $S \rightarrow \bm{v}$, we need to consider the set of all possible strings $\Omega$.
We denote the set of all category (1) strings and category (2) strings by $\cateset{1}$ and $\cateset{2}$, respectively.
By definition, $\cateset{1}$ and $\cateset{2}$ form a partition of $\Omega$.
We assume that both $\cateset{1}$ and $\cateset{2}$ are ordered so that we can query each element by its index.
Let us denote the $l$-th elements of $\cateset{1}$ and $\cateset{2}$ by $\cateset{1}[l]$ and $\cateset{2}[l]$, respectively.

Each category (1) string corresponds to an element in $\bm{v}$.

For the $i$-th category (1) string, we denote the index of the corresponding vector element in $\bm{v}$ by $\featind{i}$.
The value corresponding to this element is given by

\begin{multline}
\vecele{v}{\featind{i}} = \mbox{number of times $\cateset{1}[i]$ occurrs in $S$} , \\
\forall i \in \left\{1, 2, \ldots, \catenum{1}\right\}.
\end{multline}

We treat each media track (|trak|) segment in the metadata strings in a different way. In the MP4 file format, each |trak| node contains information about an individual track managed by the file container.
We observed that the number of tracks and content of the tracks remain consistent among devices of the same model.
The structure of an MP4 file can be better preserved if we distinguish different tracks rather than merging them.
This is achieved by assigning each track a track number in the metadata extraction phase.
For example, the first track will be |moov/trak1|, and the second track will be |moov/trak2|.
As a result, the child nodes and key-value pairs of each track will be separated, which effectively makes the feature vectors track-aware.

For category (2) strings that represent key-value pairs stored in a node, we applied a slightly different transformation strategy.

We observed that some fields in MP4 files are essentially continuous (e.g., |@avgBitrate|, |@width|).
Despite the fact that most MP4 metadata fields are discrete, assigning each combination of these continuous key-value pairs a new element in $\bm{v}$ will still result in large and redundant feature vectors.
We continue to subdivide $\cateset{2}$ based on the type of each field, where the set of strings that have discrete fields is denoted by $\cateset{2}^d$, and the set of strings that have continuous fields is denoted by $\cateset{2}^c$.
For strings that belong to $\cateset{2}^d$, the transformation scheme is similar to that of category (1) strings.
Let the vector element index of the $j$-th string in $\cateset{2}^d$ be $\featind[2][d]{j}$, then the value corresponding to the element is given by

\begin{multline}
\vecele{v}{\featind[2][d]{j}} = \mbox{number of times $\cateset{2}^d[j]$ occurs in $S$} , \\
\forall j \in \left\{1, 2, \ldots, \catenum[d]{2}\right\}.
\end{multline}

As for strings that belong to $\cateset{2}^c$, we first discard the values in the strings to form a set of distinct keys $\cateset{2}^{c^{\prime}}$, and then put the values in $\bm{v}$ directly.
For the $k$-th string in $\cateset{2}^{c^{\prime}}$, the index of the corresponding vector element in $\bm{v}$ is $\featind[2][c^{\prime}]{j}$, and the value of the element is

\begin{multline}
\vecele{v}{\featind[2][c^{\prime}]{k}} =
\begin{cases}
\mbox{the value of key $\cateset{2}^{c^{\prime}}[k]$ in $S$} \\
0~\mbox{(if the key $\cateset{2}^{c^{\prime}}[k]$ is not in $S$)}
\end{cases}
, \\
\forall j \in \left\{1, 2, \ldots, \catenum[c^{\prime}]{2}\right\}.
\end{multline}

It can be seen that the dimensionality of $\bm{v}$ is given by

\begin{align}
\dim(\bm{v}) = \catenum{1}+\catenum[d]{2}+\catenum[c^{\prime}]{2}.
\end{align}

In general, the actual value of the index functions $\adjustbox{raise=1pt,scale={1.0}{1.1}}{$\chi$}$ can be arbitrary as long as all values of $\featind{i}$, $\featind[2][d]{j}$, and $\featind[2][c^{\prime}]{k}$ form a valid partition of integers in the range $\left[1, \dim(\bm{v})\right]$.

By using different representation strategies for discrete and continuous fields (i.e., being type-aware), the resulting feature vectors are more compact and suited to machine learning techniques.

§.§ Feature Selection

Our third step consists of reducing the set of selected features by discarding redundant features.

Based on the feature extraction scheme above, it can be observed that some elements in the feature vector are significantly correlated.
For example, the presence of string |moov/mvhd/@duration=1546737| in a collection $S$ extracted from a valid MP4 file must lead to the presence of |moov/mvhd| in $S$. Therefore, feature selection can reduce redundancy within the feature vectors.

In the feature selection step, we reduce the redundancy among features that are strongly correlated.
Since only a small proportion of elements in the feature vector $\bm{v}$ are inserted from continuous fields,
most elements in $\bm{v}$ correspond to the number of occurrences of a node or a field value.
If two features in $\bm{v}$ are negatively correlated, then it often means the presence of an MP4 entity implies the absence of another MP4 entity.
In forensic scenarios, presence is much stronger evidence than absence.
Therefore, if we only focus on features that are positively correlated, then we can select features of higher forensic significance.

Given a set of feature vectors $\bm{v}_1, \bm{v}_2, \ldots, \bm{v}_N$, we can compute the corresponding correlation matrix $\bm{R}$, where $\bm{R}_{ij}$ is equal to the correlation coefficient between $i$-th and $j$-th feature in the set. Then, we set all negative elements in $\bm{R}$ to zero, which results in matrix $\bm{R}^+$.
That is, negative correlation is ignored.

Because all elements in $\bm{R}^+$ are within the range $[0,1]$, the matrix $\bm{R}^+$ can be seen as an affinity matrix between $\dim(\bm{v})$ vertices in a graph, where an affinity value of 0 indicates no connection and an affinity value of 1 indicates strongest connection.

This allow us to use spectral clustering with $\alpha$ clusters on $\bm{R}^+$, which assigns multiple strongly correlated features into the same cluster [31].
Then, we select clusters with more than $\beta$ features.

For each selected cluster, we retain only one feature at random.

Here, $\alpha > \beta > 0$ are hyperparameters.
This feature selection step helps improve the quality of feature vectors.

§.§ Dimensionality Reduction and Classification

In the last step, depending on the the video forensics problem, we use the feature vectors for classification in two ways.

Multi-class problems When the classification problem is multi-class, we use linear discriminant analysis (LDA) [11] to drastically reduce the dimensionality of feature vectors to 2 dimensions. LDA is a supervised dimensionality reduction technique.
For a classification problem of $K$ classes, LDA generates an optimal feature representation in a subspace that has dimensionality of at most $K-1$.
The new features in the subspace are optimal in the sense that the variance between projected class centroids are maximized.
For multi-class classification problems, we always reduce the dimensionality of the feature vector to 2.
After dimensionality reduction, we use a nearest neighbor classifier that uses the distance between the query sample and $\lambda$ nearest samples to make a decision, where $\lambda$ is a hyperparameter.
Each nearest sample is weighted by their distance to the query sample. The use of the dimensionality reduction and nearest neighbor classifier lead to concise and straightforward decision rules in 2D space, which can be interpreted and analyzed by human experts easily.

Two-class problems When the classification problem is two-class ($K=2$), LDA can only generate one-dimensional features. Our experiments have shown that 1D features are insufficient to represent the underlying complexity of video forensics problems. As a result, for binary classification problems, we use a decision tree classifier without dimensionality reduction.

§ EXPERIMENTS AND RESULTS

In this section we report all the details of the experiments and comment on the results.

We study the effectiveness of our approach using the following datasets:

* VISION [32]: the VISION dataset contains 629 pristine MP4/MOV videos captured by 35 different Apple, Android and Microsoft mobile devices.

* EVA-7K [39]: the EVA-7K dataset contains approximately 7000 videos captured by various mobile devices, uploaded to distinct social networks, and edited by different video manipulation tools. The authors took a subset of videos from the VISION dataset and edited them with a number of video editing tools. Then, they uploaded both the original videos and edited videos to four social networks, namely YouTube, Facebook, TikTok and WeiBo. The videos were subsequently downloaded back. The EVA-7K dataset is made up of the pristine videos, edited videos, and downloaded videos.

The VISION dataset is used for device attribution experiments, while all other experiments are conducted on the larger EVA-7K dataset.

We demonstrate the effectiveness of our approach in four video forensic scenarios.
In all of the experiments below that use LDA and nearest neighbor classification, we choose $\alpha=300$, $\beta=4$, and $\lambda = 5$.
For each of the scenarios below, unless indicated otherwise, the dataset is split into two halves with stratified sampling for training and validation, respectively.
The metadata nodes and fields that are excluded during metadata extraction, as well as the list of continuous features in the vector representation step, are provided in supplementary materials.

We mainly compare the performance of our method to [39], where the EVA-7K dataset is described.
For brand attribution, because it is conducted on the VISION dataset, we select [20] as the baseline.
The experiment results show that our approach achieves high performance evaluation metrics in all four scenarios.

§.§ Scenario 1: Brand Attribution

Brand attribution consists of identifying the brand of the device used to capture the video file under analysis.
We examined brand attribution experiments in two scenarios.
In the first experiment, we assume the analyst has access to all possible devices at training time (i.e., close-set scenario).
In the second experiment, we assume that a specific device may be unknown at training time (i.e., blind scenario).

Close-set scenario
In Table <ref>, we show the F1-score comparison between our approach and previous work.
Our method almost perfectly classifies the VISION dataset, with only one Apple device being misclassified as Samsung.
Because our framework is capable of extracting and analyzing more metadata, the performance of our method is better compared to the baseline, especially for brands like Huawei, LG, and Xiaomi.
The 2D feature distribution and decision boundary for each label are shown in Figure <ref>, from which we can determine the metadata similarity between brands and the metadata “variance” for each brand.

Visualizations as shown in Figure 6 generated by our method can aid an analyst in making a more interpretable and reliable decision.
If a new video under examination is projected close to the LG region, one can assume it is unlikely for that video to come from a Samsung device.

2D feature distribution and classification boundary for device attribution scenario.

F1-score comparison of device attribution scenario.
Device Attribution

1cBrand     Iuliani, [20]     Our Approach
Apple     1.00      0.99

Asus     1.00      1.00

Huawei      0.87     1.00

LG     0.94      1.00

Oneplus     0.93      1.00

Samsung     0.93      0.99

Wiko      0.65     1.00

Xiaomi     0.74     1.00

Blind scenario

We also examined device attribution with an unknown device using our approach.

Let us consider a specific example where device ID 20 (an Apple device) is the unknown device.
In the training phase, we use the entire VISION dataset except for samples from device ID 20.

In order to classify this device that is unknown to the classifier, we project its features in 2D space and plot them in the decision boundary plot shown in Figure <ref>.
It can be seen that all samples lie in Apple's decision region, which indicates that the unknown samples are classified correctly.

Example of blind device attribution scenario. The projected samples from the unknown device are shown with white markers with black contour.

§.§ Scenario 2: Manipulation Tool Identification

The goal of this scenario is to determine the video editing tool used to manipulate a given video file.
We considered native video files from the acquisition devices as non-edited, and compare them with video files edited using Avidemux [5], Exiftool [16], ffmpeg [10], Kdenlive [23] and Premiere [1] as reported in [39].

In Table <ref>, we compare the F1-score of our method to previous work.
Our method achieves a higher average F1-score compared to the previous work.
The 2D feature distribution and decision boundary for this scenario are shown in Figure <ref>.

F1-score comparison of manipulation tool identification scenario.
Manipulation Tool Identification

1cTool     Yang, [39]     Our Approach
Native      0.97        1.00

Avidemux      0.99      0.98

Exiftool      0.98      1.00

ffmpeg        0.94      1.00

Kdenlive      0.95      1.00

Premiere      1.00      0.99
Average     0.97       0.99

2D feature distribution and classification boundary for manipulation tool identification scenario.

§.§ Scenario 3: Social Network Attribution

In our social network attribution scenario, we classify the source social network of the video files.
If a video file is not downloaded from a social network or the video file comes from an unknown social network, it will be classified as “other”.
The F1-scores in this scenario are shown in Table <ref>; the 2D feature distribution and decision boundary for this scenario are shown in Figure <ref>.
For this task, our approach achieves high average F1-score of 0.99.
The high performance also implies that each social network leaves a unique fingerprint on its videos.

2D feature distribution and classification boundary for social network attribution scenario.

F1-score comparison of social network attribution scenario.
Social Network Attribution

1cSocial Network     Yang, [39]     Our Approach
YouTube     1.00      0.99

Facebook     1.00       1.00

WeiBo     0.99      0.99

TikTok      1.00      1.00

Other       -     0.99

§.§ Scenario 4: Manipulation Detection

There are two sub-tasks within this scenario.
As discussed above, for binary classification scenarios such as manipulation detection, LDA can only generate one-dimensional features.
It limits the features' expression power after dimensionality reduction, which leads to inferior classification performance.
Therefore, for binary classification problems, we prefer using a decision tree classifier without dimensionality reduction.

Manipulation detection on videos files from social networks

In this task, we detect manipulated videos given the fact that both pristine and edited videos are uploaded to social networks first.
We compare the performance of using our features with different classification strategies to [39], and the results are shown in Table <ref>.
Using the EVA-7K dataset, the manipulation detection problem is unbalanced because there are much more edited videos than original ones (edited:pristine$\approx$9:1), meaning more samples with positive labels.
In this case, the true positive rate (TPR) and the true negative rate (TNR) reflect a classifier's performance more objectively than F1-score.
It can be seen that our features combined with decision tree classifier achieve higher TNR for videos from all four social networks.
When we use LDA and nearest neighbor classifier for this scenario, the classifier completely fails for videos from TikTok and WeiBo.
It is likely because LDA can only generate one-dimensional features, which do not possess enough degrees of freedom to represent the complexity of this problem.
Thus, the decision tree classifier is preferred for this scenario.

From Table <ref>, we also have a glimpse of how each social network process uploaded videos.
For WeiBo and TikTok videos, conducting further metadata-based forensic analysis becomes unreliable, which indicates they may have significantly altered videos uploaded to their platforms.
YouTube videos can be classified perfectly, which implies that they apply minimum modification to videos' metadata.

Performance evaluation metrics comparison between our approach and previous work. TPR and TNR stand for True Positive Rate and True Negative Rate, respectively. The accuracy score has been balanced.
Social Network Manipulation Detection

Yang, [39]     Our Features+
Decision Tree     Our Features+
LDA & NNC
3*900.8 Facebook     Accuracy     0.76     0.84     0.62

TNR     0.40      0.87      0.30

TPR     0.86      0.82      0.95
3*900.8 TikTok     Accuracy     0.80     0.69     0.50

TNR      0.51     0.94      0.00

TPR      0.75     0.43      1.00
3*900.8 Weibo     Accuracy     0.79     0.63     0.50

TNR     0.45      0.57      0.00

TPR     0.82      0.68      1.00
3*900.8 YouTube     Accuracy     0.60     1.00     1.00

TNR     0.36      1.00      1.00

TPR     0.74      1.00      1.00

Manipulation detection on local videos

In this task, we classify pristine videos and edited videos that are not exchanged via social network.
To mimic the real world classification scenario, we employ a similar leave-one-out validation strategy as introduced in [39].
This approach takes the video files from one device model out as the validation set at a time.
Since there is only one Microsoft device among 35 models, it is discarded in this scenario, as described in [39]. Because the Microsoft device either belongs to the training set or validation set, we are left with no samples to validate or no data to train.
The mean balanced accuracy comparison of the 34-fold cross validation is shown in Table <ref>.
Our approach achieves higher performance compared to previous works.

Comparison of our method with previous works. The balanced accuracy is averaged over 34 folds.
Manipulation Detection on Local Videos

Balanced Accuracy
Güera, [15]     0.67

Iuliani, [20]     0.85

Yang, [39]     0.98

Our Features + Decision Tree     0.99

§ CONCLUSION

In this paper, we proposed a video forensics approach for MP4 video files based on metadata embedded in video files.
Our improved metadata extraction and feature representation scheme allows one to represent more metadata information in a compact feature vector.
We use feature selection, dimensionality reduction, and nearest neighbor classification techniques to form interpretable and reliable decision rules for multiple video forensics scenarios.

Our approach achieves better performance than other methods.

The performance of our method in many of the scenarios indicates that we need to increase our video forensics dataset to include more difficult cases.
Our research also exposed the limitation of metadata-based forensics methods, namely its failure to analyze videos from specific social networks such as TikTok and WeiBo.
This is a significant disadvantage compared to pixel-based methods.
In the future, we plan to continue exploring the potential of metadata-based video forensics by adding the ability to parse more manufacturer-specific data models (e.g., Canon's CNTH tags [9]) and by looking into lower-level metadata in the distribution of audio/video samples as well as location of key frames in the video stream.
We hope that metadata-based video forensics methods can be proved to be reliable in more forensic scenarios.

§ ACKNOWLEDGMENT

This material is based on research sponsored by DARPA and Air Force Research Laboratory (AFRL) under agreement number FA8750-20-2-1004.
The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.
The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA and Air Force Research Laboratory (AFRL) or the U.S. Government.
Address all correspondence to Edward J. Delp<EMAIL_ADDRESS>
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# S-PLUS: LEnticular Galaxies in Stripe 82

A. Cortesi1, K. Saha2, F.Ferrari3, G. Lucatelli3, C. Mendes de Oliveira4, S.
Dhiwar2, C. R. Bom5, L. O. Dias5
1 Observatório do Valongo, Ladeira (OV)
2 Inter-University Centre for Astronomy and Astrophysics (IUCAA)
3 Universidade Federal do Rio Grande (FURG)
4 Universidade de São Paulo
5 Brazilian Center for Physics Research (CBPF) E-mail<EMAIL_ADDRESS>
(Accepted in Anais da Academia Brasileira de Ciencias )

###### Abstract

This work is a Brazilian-Indian collaboration. It aims at investigating the
structural properties of Lenticular galaxies in the Stripe 82 using a
combination of S-PLUS (Southern Photometric Local Universe Survey) and SDSS
data. S-PLUS is a novel optical multi-wavelength survey which will cover
nearly 8000 square degrees of the Southern hemisphere in the next years and
the first data release covers the Stripe 82 area. The morphological
classification and study of the galaxies’ stellar population will be performed
combining the Bayesian Spectral type (from BPZ) and Morfometryka (MFMTK)
parameters. BPZ and MFMTK are two complementary techniques, since the first
one determines the most likely stellar population of a galaxy, in order to
obtain its photometric redshift (phot-z), and the second one recovers non-
parametric morphological quantities, such as asymmetries and concentration.
The combination of the two methods allows us to explore the correlation
between galaxies shapes (smooth, with spiral arms, etc.) and their stellar
contents (old or young population). The preliminary results, presented in this
work, show how this new data set opens a new window on our understanding of
the nearby universe.

###### keywords:

galaxies: elliptical and lenticular, cD, galaxies: individual: NGC 1023,
galaxies: kinematics and dynamics.

††pagerange: S-PLUS: LEnticular Galaxies in Stripe 82–5††pubyear: 2021

##  1 Introduction

Lenticular galaxies have been largely studied, since when they have been first
classified by Edwin Hubble Hubble (1936), as the missing link between
elliptical and spiral galaxies. S0s or lenticular galaxies present, in fact,
an hybrid structure, showing a prominent disk, but with no signs of spiral
arms. The presence of the disk suggests that their kinematics is rotationally
supported Cortesi et al. (2013a, b), as found for spiral galaxies Merrett et
al. (2003). Elliptical galaxies, on the contrary, are pressure supported
systems, as a result of their formation histories, which comprise a sequence
of major and minor mergers Bournaud, Jog & Combes (2005). Moreover, lenticular
galaxies are more frequently found in denser regions of galaxy clusters
Dressler (1980) and their frequency increases towards low redshifts Dressler
et al. (1997). Considering that this is an opposite trend to that of spiral
galaxies, it has been first hypothesised that S0 galaxies may be the end
product of spiral galaxies whose gas was gently stripped or consumed. Such
evolutionary path would require the interplay between the galaxy and the
surrounding environment to cease star formation and remove the spiral arms.
Yet, field lenticular galaxies exist. At the present moment, several
mechanisms have been pointed out as possible ways to create lenticular
galaxies: minor mergers Bournaud, Jog & Combes (2005), major mergers under
specific initial conditions Eliche-Moral et al. (2018), AGN feedback or/and
secular evolution Mishra, Wadadekar & Barway (2018), and it is becoming more
popular the idea that the S0 class is actually a compilation of objects that
formed through very different paths and just happen to have similar
appearances. Part of the problem stands on the difficulty of visually
identifying lenticular galaxies. In fact, there is not a consensus on the
morphological characteristics of lenticular galaxies and they can, in general,
be easily misclassified due to the vague appearence of spiral arms, and not
trivial determination of the presence of a disk, from a purely photometric
approach. Large spectroscopic surveys, on the other side, often covers only
the central part of a galaxy, i.e. the region where the bulge dominates, again
making it hard to separate elliptical galaxies, from lenticular galaxies with
large bulge-over-total light ratio. Multi-wavelength surveys (as S-PLUS Mendes
de Oliveira et al. (2019)) open a new window on galaxy classification,
combining the possibility of studying the variation of morphological
parameters with wavelengths and recovering the galaxy stellar population. We
present preliminary results of this approach, using the S-PLUS first data
release111https://datalab.noao.edu/splus/, and we describe new methods to
identify lenticular galaxies.

##  2 Data and Methods

S-PLUS (Southern Photometric Local Universe Survey) is an astronomical
facility in Chile (Cerro Pachon), dedicated to mapping the observable sky in 7
narrow-band filters and 5 broad-band (ugriz) filters in the optical region.
The 0.86m mirror of the T80-South telescope, combined with a field of view of
1.4 square degrees and an 85 Mega-pixel camera, is producing high-quality
images and a unique spectral resolution for millions of objects over several
thousand square degrees. Together with its twin observatory in the northern
hemisphere, the T80-North, its sister survey J-PLUS, will cast the first light
on a multi-colour Universe, on about half of the extragalactic sky. This
multi-purpose astrophysical survey in the southern hemisphere started in 2016.
During the next 3-4 years it will observe more than 8,000 square degrees (1/5
of the whole sky), covering the entire visible region of the electromagnetic
spectrum (3500 A to 10,000 A). Figure 1 and 2 show the area covered in the sky
and the Javalambre filter system used by S-PLUS.

Galaxy morphologies vary with wavelength, since different stellar populations
present different colours. In general, star forming galaxies have a lower
bulge-to-total ($B/T$) light ratio than quiescent ones Morselli et al. (2017).
We use Morfometryka Ferrari, de Carvalho & Trevisan (2015) (MFMTK) to compute
galaxies’ non-parametric morphometric parameters, in particular we recover the
concentration (C1) and the entropy (H). The concentration is defined as the
ratio of the radii containing some fraction of the total light inside the
Petrosian Region (2Rp):

$C_{1}=\log\left(\frac{R_{80}}{R_{20}}\right)\qquad,$

where $R_{f}$ is the radius that contains a fraction of $f$ %. Entropy is
defined as:

$H(I)=-\sum_{i}^{n}p(I_{i})\ln[p(I_{i})]/H_{max},$

where $p$ is the probability of the occurence of the intensity $I_{i}$ and
$H_{max}$ is the maximum entropy, which is for an homogeneous distribution
(e.g. $p=1/n$). For the early type galaxies the concentration is nearly
constant with wavelength, while it increases for redder wavelengths, in the
case of the spiral galaxies. Another feature is that homogeneous systems (with
less concentrated light distributions, such as disks or extended galaxies)
have higher entropies than systems with more concentrated light distributions
(ellipticals, bulge dominated, compacted, etc.). Finally, we use the Bayesian
Photometric redshifts (BPZ) Molino et al. (2014) method to obtain the galaxy
bayesian spectral type, by identifying the galaxy template model that
optimises the redshift determination, for more details see Molino et al.
(2020).

Figure 1: Diagram showing some of the most important optical and near-infrared
surveys in the Southern Hemisphere (we omit the surveys SkyMapper, Gaia and
LSST that cover the entire hemisphere or sky). For the optical surveys: ATLAS
is shown in hatched green, VPHAS+ is the pink rectangular contour over the
Bulge and Disk of the Galaxy, DECAL is in hatched black, DES is shown in blue,
KiDS in yellow, and GAMA in bright green. The only near-infrared survey
displayed is VISTA-VVV, a rectangle with light gray contours over the Galactic
disk. Dark red shows the area covered by S-PLUS. The broken black line
represents the ecliptic. The figure is taken from Mendes de Oliveira et al.
(2019). Figure 2: The Javalambre 12-filter system used by S-PLUS. The y-axis
shows the total efficiency of the filters, obtained through the multiplication
of the average transmission curves, the CCD efficiency, and the mirror
reflectivity curves. Different filters are coloured according to the labels on
the right-side panel. The figure is taken from Mendes de Oliveira et al.
(2019).

##  3 Results

The top panel of Figure 3 shows the distribution of the galaxies in the $H-C1$
plane, colour coded according to their morphological type Nair & Abraham
(2010). The T-Type parameter value is lower than $-3$ for elliptical galaxies,
between $-3$ to $0$ for lenticular galaxies, and higher than $0$ for spiral
and irregular galaxies. This plot clearly shows that only using these two
MFMTK Ferrari, de Carvalho & Trevisan (2015) parameters we are able to
separate galaxies into different classes. Combining these parameters with
other MFMTK outputs (such as asymmetries, clumpiness, Gini parameter,
spirality ..) and the curvature of the brightness profile of the galaxies
Lucatelli & Ferrari (2019) at every wavelengths, we are able to identify a
bona-fide catalogue of lenticular galaxies for the S-PLUS DR1 data release
(Lucatelli et al. in prep).

The bottom panel of Figure 3 presents the same $H-C1$ relation, but this time
the objects are colour coded according to their Bayesian Spectral Type (Tb),
as defined in Molino et al. (2020). The galaxies’ templates are defined as
follows: from T1 to T4 is a spectral energy distribution (SED) typical of
elliptical galaxies, T5 is for ES0 galaxies, from T6 to T14 are spiral
galaxies templates (barred and un-barred). Again, the morphometric parameters
allow us to divide galaxies into passive (early-type galaxies) and star
forming (late-type galaxies). Interestingly, several T1 galaxies lie in the
part of the plot populated by spiral galaxies, and some galaxies characterised
by a spiral-like galaxy templates fall in the upper area of the plot, where we
expect to find early type galaxies. This finding shows that galaxies’
morphology and stellar content are not always as expected: the universe
present blue ellipticals and red spirals Strateva et al. (2001). S-PLUS data
set allows to recover both these quantities and map the galaxy distribution in
the local universe.

Figure 3: Entropy versus concentration parameters, estimated using MFMTK, for
a sample of 925 galaxies, colour coded according to their morphological
T-Type, top panel, and according to their bayesian spectral templates, bottom
panel. See text for more details.

##  4 Conclusions

In this work we present preliminary results of the study of galaxy morphology
and stellar population properties obtained using data of the S-PLUS survey. We
show how the parameters recovered with MFMTK allow to perform a galaxy
morphological classification and we suggest a way to identify S0 galaxies,
combining such parameters with a novel method based on the study of the
curvature of the galaxies’ surface light profile Lucatelli & Ferrari (2019).
The S-PLUS data set, given its unique combination of narrow and broad band
filters, allows the estimation of the galaxy most probable stellar template,
together with the definition of its morphometric parameters. This study
extends from observational astronomy, to theory and computational analysis; it
comprises researchers from Brazil and India and it is open to a world-wide
collaboration. With its 0.86 cm diameter, S-PLUS proves the importance of
small telescopes in the advance of astronomical knowledge. Several other small
telescopes exist in the BRICS countries. We conclude encouraging a dialogue
among these telescope users, sharing technical and scientific knowledge.

##  5 Acknowledgments

AC acknowledges support from PNPD/CAPES.

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ĐặngVõ Phúc Faculty of Education Studies, University of Khanh Hoa, Vietnam
Primary cohomology operationsSteenrod algebraHit problemCohomology of Steenrod
algebraAlgebraic transferLambda algebraAdams spectral sequence

# On the lambda algebra and Singer’s cohomological transfer

###### Abstract

Writing $\mathbb{A}$ for the 2-primary Steenrod algebra, which is the algebra
of stable natural endomorphisms of the mod 2 cohomology functor on topological
spaces. Working at the prime 2, computing the cohomology of $\mathbb{A}$ is an
important problem of Algebraic topology, because it is the initial page of the
Adams spectral sequence converging to stable homotopy groups of the spheres. A
relatively efficient tool to describe this cohomology is the Singer algebraic
transfer of rank $n$ in [22], which passes from a certain subquotient of a
divided power algebra to the cohomology of $\mathbb{A}.$ Singer predicted that
this transfer is a monomorphism, but this remains open for $n\geq 4.$ This
short note is to verify the conjecture in the ranks 4 and 5 and some generic
degrees.

##  1 Introduction

It is well-known that there is a group homomorphism $Sq^{n}$ for $n\geq 0,$
between mod-2 cohomology groups of a topological space, called Steenrod
squares of degrees $n$. They are stable cohomology operations, that is, they
commute with suspension maps. All the Steenrod squares form an algebraic
structure which is known as the 2-primary Steenrod algebra $\mathbb{A}$
subject to the Adem relations. It was applied to the vector fields on spheres
and the Hopf invariant one problem, which asks for which $n$ there exist maps
of Hopf invariant $\pm 1$. So, the Steenrod algebra is one of the important
tools in Algebraic topology. Specifically, its cohomology ${\rm
Ext}_{\mathbb{A}}^{*,*}(\mathbb{Z}/2,\mathbb{Z}/2)$ is an algebraic object
that serves as the input to the Adams spectral sequence (the ASS) [1] and
therefore, computing this cohomology is of fundamental importance to the study
of the stable homotopy groups of spheres.

The cohomological transfer, defined by Singer [22], could be an useful
approach to describe the mysterious structure of the cohomology algebra of
$\mathbb{A}$. In order to better understand this transfer, we will use the
following notations and the relevant concepts. Let denote $V^{\oplus n}$ the
$n$-dimensional vector space over the prime field $\mathbb{Z}/2.$ Then, we
write $H^{*}(V^{\oplus n})$ and $H_{*}(V^{\oplus n})$ for mod-2 cohomology and
homology of $BV^{\oplus n}$ (the classifying space of $V^{\oplus n}$). One
should note that $BV^{\oplus n}$ is homotopy equivalent to the cartesian
product of $n$ copies of the union of the finite projective spaces. As it is
known, $H^{*}(V^{\oplus n})$ is identified with the polynomial algebra
$\mathbb{Z}/2[u_{1},\ldots,u_{n}]$ on generators of degree $1,$ equipped with
the canonical unstable algebra structure over the Steenrod algebra (i.e., it
is a commutative, associative, graded $\mathbb{Z}/2$-algebra equipped with a
structure of unstable $\mathbb{A}$-module and satisfying two relations, one
called the Cartan formula, and the other called the instability relation:
$Sq^{\deg(x)}(x)=x^{2}$.) By dualizing, $H_{*}(V^{\oplus n})$ has a natural
basis dual to the monomial basis of $H^{*}(V^{\oplus n}).$ We denote by
$a_{1},\ldots,a_{n}$ the basis of $H_{1}(V^{\oplus n})$ dual to the basis
$u_{1},\ldots,u_{n}$ of $H^{1}(V^{\oplus n})={\rm Hom}(V^{\oplus
n},\mathbb{Z}/2),$ so that $a_{i}(u_{j})=1$ if $i=j$ and is $0$ if $i\neq j.$
We write the dual of $u_{1}^{d_{1}}\ldots u_{n}^{d_{n}}$ as
$a_{1}^{(d_{1})}\ldots a_{n}^{(d_{n})}$ where the parenthesized exponents are
called divided powers, and be careful that in the corresponding situation over
a field of characteristic $0$ in place of $\mathbb{Z}/2,$
$a_{i}^{(d_{i})}=a_{i}^{d}/d!,$ which fits with the formula
$a^{(d)}a^{(e)}=\binom{d+e}{d}a^{(d+e)}.$ This product gives a commutative
graded algebra $\Gamma(a_{1},\ldots,a_{n})$ over $\mathbb{Z}/2$ called a
divided power algebra, where $\Gamma(a_{1},\ldots,a_{n})=H_{*}(V^{\oplus n}),$
and an element $a_{1}^{(d_{1})}\ldots a_{n}^{(d_{n})}$ in $H_{*}(V^{\oplus
n})$ corresponding to a monomial $u_{1}^{d_{1}}\ldots u_{n}^{d_{n}}$ in
$H^{*}(V^{\oplus n})$ is called $d$-monomial. Now, let
$P_{\mathbb{A}}H_{*}(V^{\oplus n})$ be the subspace of $H_{*}(V^{\oplus n})$
consisting of all elements that are annihilated by all Steenrod squares of
positive degrees. The general linear group $GL_{n}=GL(V^{\oplus n})$ acts
regularly on the classifying space $BV^{\oplus n}$ and therefore on
$H^{*}(V^{\oplus n})$ and $H_{*}(V^{\oplus n}).$ This action commutes with
that of the algebra $\mathbb{A}$ and so acts
$\mathbb{Z}/2\otimes_{\mathbb{A}}H^{*}(V^{\oplus n})$ and
$P_{\mathbb{A}}H_{*}(V^{\oplus n}).$ For each $n\geq 0,$ Singer constructed in
[22] a linear transformation from $P_{\mathbb{A}}H_{*}(V^{\oplus n})$ to the
$n$-th cohomology group ${\rm
Ext}_{\mathbb{A}}^{n,n+*}(\mathbb{Z}/2,\mathbb{Z}/2)$ of $\mathbb{A}$, which
commutes with two $Sq^{0}$’s on $P_{\mathbb{A}}H_{*}(V^{\oplus n})$ and ${\rm
Ext}_{\mathbb{A}}^{n,n+*}(\mathbb{Z}/2,\mathbb{Z}/2)$ (see Boardman [2] and
Minami [13] for more about this). He shows that this map factors through the
quotient of its domain’s $GL_{n}$-coinvariants to give rise the so-called
cohomological transfer of rank n
$\begin{array}[]{ll}\phi_{n}^{*}(\mathbb{Z}/2)&:\mathbb{Z}/2\otimes_{GL_{n}}P_{\mathbb{A}}H_{*}(V^{\oplus
n})\\\ &\hskip 82.51282pt\longrightarrow{\rm
Ext}_{\mathbb{A}}^{n,n+*}(\mathbb{Z}/2,\mathbb{Z}/2).\end{array}$
The domain of this transfer is dual to the space of $GL_{n}$-invariants
$(\mathbb{Z}/2\otimes_{\mathbb{A}}H^{*}(V^{\oplus n}))^{GL_{n}}.$ It is to be
noted that $\phi_{n}^{*}(\mathbb{Z}/2)$ is induced over the $E_{2}$-term of
the ASS by the geometrical transfer map $\Sigma^{\infty}(B(V^{\oplus
n})_{+})\longrightarrow\Sigma^{\infty}(\mathbb{S}^{0})$ in stable homotopy
theory (see also Mitchell [14]). The work of Minami [13] indicated that these
transfers play a key role in finding permanent cycles in the ASS. In the
second cohomology groups of $\mathbb{A}$, following Mahowald [11] and Lin-
Mahowald [9], the classes $h_{1}h_{j}$ for $j\geq 3$ and $h_{j}^{2}$ for
$0\leq j\leq 5,$ are known to be the permanent cycles in the ASS. In 2016,
Hill, Hopkins, and Ravenel [6] showed that when $j\geq 7,$ the class
$h_{j}^{2}$ is not a permanent cycle in the ASS. It is surprising that so far
there is no answer for $j=6.$ The question of whether these $h_{j}^{2}$ are
the permanent cycles in the ASS or not is called Kervaire invariant problem in
literature [4]. This is one of the oldest unresolved issues in Differential
and Algebraic topology.

Direct calculating the value of $\phi_{n}^{*}(\mathbb{Z}/2)$ on any non-zero
element is a hard work. It has been demonstrated that
$\phi_{n}^{*}(\mathbb{Z}/2)$ is an isomorphism for $n\leq 2$ by Singer himself
[22], and $n=3$ by Boardman [2]. Most notably, Singer sets up a hypothesis in
the same paper [22] that $\phi_{n}^{*}(\mathbb{Z}/2)$ is a monomorphism, but
this is still not confirmed, for all cohomological degrees $n\geq 4.$ The
cases $n=4,\,5$ are our concern in this paper. Besides Singer’s transfer
homomorphism, the lambda algebra $\Lambda$ of Bousfield et al. [3] is also a
relatively efficient tool to compute the cohomology of the Steenrod algebra.
Recall that $\Lambda$ is the quotient of the graded tensor algebra over
$\mathbb{Z}/2$ on symbols $\lambda_{i}$ for $i\geq-1,$ modulo the two-sided
ideal generated by
$\lambda_{s}\lambda_{k}-\sum_{j}\binom{j-k-1}{2j-s}\lambda_{s+k-j}\lambda_{j},$
for any $s,\,k\geq-1$ by the right ideal generated by $\lambda_{-1}.$ An
interesting representation in the algebra $\Lambda$ of the algebraic transfer,
established by Chơn and Hà [5], is a $\mathbb{Z}/2$-linear map $\psi_{n}$ from
$P_{\mathbb{A}}H_{*}(V^{\oplus n})$ to a subspace of $\Lambda$ spanned by all
monomials of length $n$ in all the monomials in $\lambda_{i}.$ The authors
showed that the image of an element $\zeta\in P_{\mathbb{A}}H_{*}(V^{\oplus
n})$ under $\psi_{n}$ is a cycle in $\Lambda$ and
$[\psi_{n}(\zeta)]=\phi_{n}^{*}(\mathbb{Z}/2)([\zeta]).$ Note also that this
result is a dual version of the one in Hưng [7].

The Singer transfer we are discussing is closely related to the hit problem in
literature [16] of determination of a minimal generating set for the unstable
$\mathbb{A}$-module $H^{*}(V^{\oplus n}).$ The reader can find an excellent
list of publications about this problem in the works by Kameko [8], Mothebe-
Uys [15], the present writer [17, 18, 21], Singer [23], Sum [24, 26], Walker-
Wood [28], Wood [29] and others. Hit problems are motivated by several
problems in Topology and Algebra. It was completely studied for the cases
$n\leq 4$ by Peterson [16], Kameko’s thesis [8] and Sum [24]. Nevertheless,
the general answer seems to be out of reach with the present techniques.
Therefore, it is renowned as a difficult problem, even with the help of a
computer. In fact, when $\mathbb{Z}/2$ is a trivial $\mathbb{A}$-module,
solving the hit problem is equivalent to determining the ”cohits”
$\mathbb{Z}/2\otimes_{\mathbb{A}}H^{*}(V^{\oplus n})$ as a graded vector
space, or more generally as a graded module over the group algebra
$\mathbb{Z}/2[GL_{n}].$ Frank Peterson [16] conjectured that
$\mathbb{Z}/2\otimes_{\mathbb{A}}H^{*}(V^{\oplus n})=0$ unless
$\alpha(n+*)\leq n,$ where $\alpha(k)$ is the number of $1$’s in the dyadic
expansion of a positive integer $k.$ His motivation for this was to prove that
if $\mathcal{M}$ is a smooth manifold of dimension $*$ such that all products
of length $n$ of Stiefel-Whiney classes of its nomal bundle vanish, then
either $\alpha(*)\leq n$ or $\mathcal{M}$ is cobordant to zero. The conjecture
was established by Wood [29]. Therefrom, to study
$\mathbb{Z}/2\otimes_{\mathbb{A}}H^{*}(V^{\oplus n})$ in each $n$ and degree
$*\geq 0$, it suffices by Peterson’s conjecture and iteration of the Kameko
map [8] to consider degrees $*$ in the following ”generic” form:

(1) $k(2^{t}-1)+\ell.2^{t},\ \mbox{ for $k,\,t,\,\ell\geq 0,\,\mu(\ell)<k\leq
n$},$

where $\mu(\ell)$ is a smallest number $r\in\mathbb{N}$ such that
$\alpha(\ell+r)\leq r.$ By Minami [12], hit problems are also considered as an
useful tool for studying permanent cycles in the ASS.

In the present work, we explicitly determine the structure of the coinvariant
$\mathbb{Z}/2\otimes_{GL_{n}}P_{\mathbb{A}}H_{*}(V^{\oplus n})$ and the
behavior of the cohomological transfer of ranks 4 and 5 in some generic
degrees of the form (1) by using techniques of the hit problem and the
representation in the algebra $\Lambda$ of these transfers.

##  2 Main results

To begin with, we remark that by Sum [24], it is enough to depict the behavior
of the fourth transfer in the following degrees $d$:
$\begin{array}[]{lll}{\rm(i)}&d&=2^{t+1}-r,\ \mbox{for $1\leq r\leq 3,$}\\\
{\rm(ii)}&d&=2^{t+s+1}+2^{t+1}-3,\\\ {\rm(iii)}&d&=2^{t+s}+2^{t}-2,\\\
{\rm(iv)}&d&=2^{t+s+u}+2^{t+s}+2^{t}-3,\end{array}$
whenever $r,\,t,\,s,$ and $u$ are positive integers. It is not difficult to
check that the above degrees can be rewritten as (1). The cases of (i) are
known by Sum [25]. The results for (ii) and (iii) have been partially probed
in [19, 20]. This note is to investigate the case (iv).

Now, let us state the main results of this text. We first study the fourth
cohomological transfer in degrees $d_{t,\,s,\,u}:=2^{t+s+u}+2^{t+s}+2^{t}-3$.
To make this, we give an explicit description of a minimal generating set for
the domain of $\phi_{4}^{*}(\mathbb{Z}/2)$ in degree $d_{t,\,s,\,u},$ and
obtain the following, which is proved in many steps by using computational
techniques of the hit problem of four variables and some previous results by
the present author [19, 20] and Sum [25].

######  Theorem 2.1.

The domain of $\phi_{4}^{*}(\mathbb{Z}/2)$ in degree $d_{t,\,s,\,u}$ is
determined by

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\mathbb{Z}/2\otimes_{GL_{4}}P_{\mathbb{A}}H_{d_{t,\,s,\,u}}(V^{\oplus
4})\\\ &=\left\\{\begin{array}[]{ll}0&\mbox{if $s=1$, $u=1$ and $t\geq 1$},\\\
0&\mbox{if $s=2$, $u=1$ and $t\geq 2$},\\\ 0&\mbox{if $s\geq 3$, $u=1$ and
$t\geq 1$},\\\ 0&\mbox{if $s=1$, $u=2$ and $t=1$},\\\
\langle[\zeta_{t,\,1,\,2}]\rangle,&\mbox{if $s=1$, $u=2$ and $t\geq 2$},\\\
0&\mbox{if $s=1$, $u\geq 3$ and $t\geq 1$},\\\
\langle[\zeta_{1,\,2,\,u}]\rangle,&\mbox{if $s=2$, $u\geq 1$ and $t=1$},\\\
0&\mbox{if $s\geq 3$, $u\geq 2$ and $t=1$},\\\
\langle[\zeta_{t,\,s,\,u}]\rangle,&\mbox{if $s\geq 2$, $u\geq 2$ and $t\geq
2$},\\\ \end{array}\right.\end{array}$

where
$\begin{array}[]{ll}\zeta_{t,\,1,\,2}&=a_{1}^{(0)}a_{2}^{(2^{t+2}-1)}a_{3}^{(2^{t+2}-1)}a_{4}^{(3.2^{t}-1)}\\\
&\quad+a_{1}^{(0)}a_{2}^{(2^{t+2}-1)}a_{3}^{(5.2^{t}-1)}a_{4}^{(2^{t+1}-1)}\\\
&\quad+a_{1}^{(0)}a_{2}^{(6.2^{t}-1)}a_{3}^{(3.2^{t}-1)}a_{4}^{(2^{t+1}-1)}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+a_{1}^{(0)}a_{2}^{(7.2^{t}-1)}a_{3}^{(2^{t+1}-1)}a_{4}^{(2^{t+1}-1)},\\\
\zeta_{1,\,2,\,u}&=a_{1}^{(2^{u+3}-1)}a_{2}^{(3)}a_{3}^{(3)}a_{4}^{(2)}\\\
&\quad+a_{1}^{(2^{u+3}-1)}a_{2}^{(3)}a_{3}^{(4)}a_{4}^{(1)}\\\
&\quad+a_{1}^{(2^{u+3}-1)}a_{2}^{(5)}a_{3}^{(2)}a_{4}^{(1)}\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt&\quad+a_{1}^{(2^{u+3}-1)}a_{2}^{(6)}a_{3}^{(1)}a_{4}^{(1)},\\\
\zeta_{t,\,s,\,u}&=a_{1}^{(0)}a_{2}^{(2^{t}-1)}a_{3}^{(2^{s+t}-1)}a_{4}^{(2^{s+t+u}-1)}.\end{array}$

It would be interesting also to see that by this theorem, we have
isomorphisms:
$\mathbb{Z}/2\otimes_{GL_{4}}P_{\mathbb{A}}H_{d_{t,\,1,\,2}}(V^{\oplus
4})\cong\mathbb{Z}/2,\ (t\geq 2),\\\
\mathbb{Z}/2\otimes_{GL_{4}}P_{\mathbb{A}}H_{d_{1,\,2,\,u}}(V^{\oplus
4})\cong\mathbb{Z}/2,\ (u\geq 1),\\\
\mathbb{Z}/2\otimes_{GL_{4}}P_{\mathbb{A}}H_{d_{t,\,s,\,u}}(V^{\oplus
4})\cong\mathbb{Z}/2,\,(t\geq 2,s\geq 2,u\geq 2).$
Since the elements $\zeta_{t,\,1,\,2},$ $\zeta_{1,\,2,\,u}$ and
$\zeta_{t,\,s,\,u}$ belong to $P_{\mathbb{A}}H_{d_{t,\,1,\,2}}(V^{\oplus 4}),$
$P_{\mathbb{A}}H_{d_{1,\,2,\,u}}(V^{\oplus 4})$ and
$P_{\mathbb{A}}H_{d_{t,\,s,\,u}}(V^{\oplus 4}),$ respectively,
$\psi_{4}(\zeta_{t,\,1,\,2}),$ $\psi_{4}(\zeta_{1,\,2,\,u})$ and
$\psi_{4}(\zeta_{t,\,s,\,u})$ are cycles in the lambda algebra $\Lambda$.
Furthermore, using the representation in $\Lambda$ of the fourth transfer map,
it may be concluded that
$\begin{array}[]{ll}&\phi_{4}^{*}(\mathbb{Z}/2)([\zeta_{t,\,1,\,2}])=[\psi_{4}(\zeta_{t,\,1,\,2})]\\\
&=[\lambda_{0}\lambda_{2^{t+2}-1}^{2}\lambda_{2^{t+1}+2^{t}-1}]\\\
&=h_{0}c_{t}\in{\rm
Ext}_{\mathbb{A}}^{4,4+d_{t,\,1,\,2}}(\mathbb{Z}/2,\mathbb{Z}/2),\\\
&\phi_{4}^{*}(\mathbb{Z}/2)([\zeta_{1,\,2,\,u}])=[\psi_{4}(\zeta_{1,\,2,\,u})]\\\
&=[\lambda_{2^{u+3}-1}\lambda_{3}^{2}\lambda_{2}]\\\ &=h_{u+3}c_{0}\in{\rm
Ext}_{\mathbb{A}}^{4,4+d_{1,\,2,\,u}}(\mathbb{Z}/2,\mathbb{Z}/2),\\\
&\phi_{4}^{*}(\mathbb{Z}/2)([\zeta_{t,\,s,\,u}])=[\psi_{4}(\zeta_{t,\,s,\,u})]\\\
&=[\lambda_{0}\lambda_{2^{t}-1}\lambda_{2^{s+t}-1}\lambda_{2^{s+t+u}-1}]\\\
&=h_{0}h_{t}h_{s+t}h_{s+t+u}\in{\rm
Ext}_{\mathbb{A}}^{4,4+d_{t,\,s,\,u}}(\mathbb{Z}/2,\mathbb{Z}/2).\end{array}$
These, together with Theorem 2.1 and a fact of the fourth cohomology groups
${\rm Ext}_{\mathbb{A}}^{4,4+d_{t,\,s,\,u}}(\mathbb{Z}/2,\mathbb{Z}/2)$ (see
Lin [10]), yields the following.

######  Corollary 2.2.

The algebraic transfer is an isomorphism in bidegree $(4,4+d_{t,\,s,\,u})$ for
all $t,\,s,\,u.$

Next, we are going to survey the behavior of the transfer homomorphism of rank
5 in degrees of the form (1) with $k=n=5$ and $\ell=50.$ To accomplish this,
the domains of $\phi_{5}^{*}(\mathbb{Z}/2)$ at these degrees are computed as
follows.

######  Theorem 2.3.

Let $d_{t}:=5(2^{t}-1)+50.2^{t}$ with $t$ an arbitrary non-negative integer.
Then, the coinvariant spaces
$\mathbb{Z}/2\otimes_{GL_{5}}P_{\mathbb{A}}H_{d_{t}}(V^{\oplus 5})$ are
trivial.

The proof of the theorem is quite long and complicated. We give a brief
description for this: Firstly, we compute explicitly the monomial bases for
the spaces of indecomposable elements
$\mathbb{Z}/2\otimes_{\mathbb{A}}H^{*}(V^{\oplus 5})$ in degrees $d_{t}.$ An
effective approach, based on the iterated Kameko squaring operation [8]
coupled with the results by Mothebe-Uys [15], Tín [27] and some linear
transformations in Sum [24], is applied in our computations Additionally, we
also used the MAGMA computer algebra to verify the results. In particular,
these calculations confirmed Sum’s conjecture [26] for the relationship
between $\mathbb{A}$-generators of the algebras $H^{*}(V^{\oplus 4})$ and
$H^{*}(V^{\oplus 5})$ in the above degrees. Next, using those monomial bases,
we may claim that the invariants
$(\mathbb{Z}/2\otimes_{\mathbb{A}}H^{*}(V^{\oplus 5}))^{GL_{5}}$ are trivial
in degrees $d_{t},$ from which the theorem follows from the fact that
$\mathbb{Z}/2\otimes_{GL_{5}}P_{\mathbb{A}}H_{*}(V^{\oplus 5})$ is isomorphic
to $(\mathbb{Z}/2\otimes_{\mathbb{A}}H^{*}(V^{\oplus 5}))^{GL_{5}}.$

Now according to [10], it is easily seen that
${\rm Ext}_{\mathbb{A}}^{5,5+d_{t}}(\mathbb{Z}/2,\mathbb{Z}/2)=\langle
h_{0}h_{t+1}h_{t+2}h_{t+4}h_{t+5}\rangle=0,$ for arbitrary $t\geq 0,$ and so,
by Theorem 2.3, we immediately obtain

######  Corollary 2.4.

The Singer transfer is a trivial isomorphism in bidegree $(5,5+d_{t})$ for
every non-negative integer $t.$

Thus, Corollaries 2.2 and 2.4 favor the Singer conjecture in bidegrees
$(4,4+d_{t,\,s,\,u})$ and $(5,5+d_{t})$ for any $t,\,s,\,u.$

Detailed proofs of all the results of this note will be published elsewhere.

##  3 Conclusion

Although our work does not apparently lead to either a proof or a refutation
of the Singer conjecture in general, we feel that it represents an interesting
note about an application of hit problem and the lambda algebra for studying
this conjecture. Perhaps a continuation of our methods may provefruitful. It
is our belief that further research canspring from these ideas.

######  Acknowledgments.

This work belongs to the Research Project 2022. The author would like to
express warm gratitude to Prof. Nguyễn Sum for many enlightening e-mail
exchanges.

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