I Introduction
Many machine learning and modern signal processing applications
such as biometric authentication/identification and recommending systems , follow sparse signal processing techniques [1, 2, 3, 4, 5, 6]. The sparse synthesis model focuses on those data sets that can be approximated using a linear combination of only a small number of cells of a dictionary. The applications of sparse coding based dictionary learning go chiefly over either or both extraction and estimation of local features. Typically, this kind of discipline is controlled via a prior decomposition of the original signal into overlapping blocks
call sideinformation. In relation to this strategy, there mainly exist two demerits: (i) each point in the signal is estimated multiple times something that shows a redundancy; and (ii) a shifted version of the features interested to be learnt are captured since the correlations/sideinformation among neighboring data sets are not fully taken into account something that results in the inevitable fact that some portions of the learnt dictionaries may be practically of a partially useless nature.A quick overview over the literature in terms of keygeneration and authentication is strongly widely provided here in details. In [7] and for cloudassisted autonomous vehicles, Q. Jiang et al technically proposed a biometric privacy preserving threefactor authentication and key agreement. In [8] and through vehicular crowdsourcing, F. Song et al technically proposed a privacypreserving task matching with threshold similarity search. In [9] and for mobile InternetofThings (IoT), X. Zeng, et al technically proposed a highly effective anonymous user authentication protocol. In [10] and for fogassisted IoT, J. Zhang et al technically proposed a revocable and privacypreserving decentralized data sharing scheme. In [11] and for resourceconstrained IoT devices, Q. Hu et al technically proposed a halfduplex mode based secure key generation method. In [12] and for mobile crowdsensing under untrusted platform, Z. Wang et al theoretically explored privacydriven truthful incentives. In [13] and for IoTenabled probabilistic smart contracts, N. S. Patel et al theoretically explored blockchainenvisioned trusted random estimators. In [14] and for IoT, R. Chaudhary et al theoretically explored Latticebased public key cryptomechanisms. In [15] and for vehicle to vehicle communication, V. Hassija et al
theoretically explored a scheme with the aid of directed acyclic graph and game theory. In
[16] and for vehicular social networks, J. Sun et al theoretically explored a secure flexible and tamperingresistant data sharing mechanism. In [17] and for missioncritical IoT applications, H. Wang et al theoretically explored a secure shortpacket communications. In [18] and for edgeassisted IoT, P. Gope et al theoretically proposed a highly effective privacypreserving authenticated key agreement framework.Overall, many works have been fulfilled in the context of secret keygeneration, however, sparse coding and an optimised problem in this area has not been introduced so far. In addition, although the literature has tried its best to propose novel frameworks [7, 9, 10, 13, 16, 17, 18] as well as the optimisation techniques [8, 11, 12, 14, 15] and the relative relaxations, the population of the research and work done in this context still essentially lacks and requires to be further enhanced.
Ia Motivations and contributions
In this paper, we are interested in responding to the following question: How can we guarantee highly adequate relaxations over a ratedistortion based sparse dictionary learning? How is Alice able to efficiently utilise totally disparate resources to guarantee her private message to be concealed from Eve? With regard to the noncomplete version of the literature, the expressed question strongly motivate us to find an interesting solution, according to which our contributions are fundamentally described as follows.

(i) We initially write a ratedistortion based sparse coding optimisation problem. Subsequently, we try to add some extra constraints which are of a purely meaningful nature in this context.

(ii) We show that we can get access to the time evolution of the DegreesofFreedom (DoF) in which we are interested. We do this through a Langevin equation.

(iii) We then add an additional constraint in relation to the sideinformation.

(iv) Penultimately, we make trials to theoretically relax the resultant optimisation problem w.r.t. the facts of Graphon and a stochastic Chordal SchrammLoewner evolution etc. something that can be chiefly elaborated according to some theoretical principles such as Riemannian gradients in Riemannian geometry and the Kroner’s graph entropy.

(v) Finally, we also extend our scenario to the eavesdropping case. We in fact technically define some probabilistic constraints to be added to the main optimisation problem.
IB General notation
The notations widely used throughout the paper is given in Table I.
Notation  Definition  Notation  Definition 

Problem  Problem  
Problem  New Version of Problem  
Perturbation  Expectedvalue  
Time Instance  Sideinformation  
Distributed  Randomwalk  
Specific Time Instance  Arbitary Threshold  
Distortion Function  Mutual Information  
Graph  Brownian Motion 
IC Organisation
The rest of the paper is organised as follows. The system setup and our main results are given in Sections II and III. Subsequently, the evaluation of the framework and conclusions are given in Sections IV and V.
Ii System model and problem formulation
In this section, we describe the system model, subsequently, we formulate the basis of our problem.
Iia Flow of the problemandsolution
Initially speaking, take a quick look at Fig. 1. It basically says:

ProblemandSolution is to find the minimal set which guarantees a ratedistortion tradeoff;

DoF stands fundamentally theoretically for the minimal set expressed above which essentially talk about the transition jumps;

Tradeoff is the ratedistortion problem;

Reason of sideinformation and the logic behind of using that is to control the sparse coding^{1}^{1}1Something that is traditional as mentioned in the paper in the next parts.;

Sparse secretkey generation and the logic behind of using that is to guarantee a correlated minimal set available between Bob and Eve;

Time evolution of DoF is explored via a Langevin equation;

Relaxation of the ratedistortion problem is interpreted through the Graphon principle and w.r.t. a stochastic Chordal SchrammLoewner evolution; and

Logic of time in optimisation is to guarantee an instantaneous optimisation.
IiB System description
A traditional sparse coding scheme practically includes the following steps [1]:

Sender: With the aid of a sparsifying transform which should be trained to the receiver call Bob via the server, the sender call Alice generates the sparse codewords from her own data. She subsequently shares the privacyprotected sparse codebook with the the server as the service provider. Alice now decides^{2}^{2}2According to the Kerckchoffs’s principle in cryptography [1]. to permit a licensed availability, i.e., publicity over the sparsifying transform learnt by the client(s).

Server Step 1 as an indexing for the sender: The server marks some indexes on the received sparsecodes in a database.

Client: Via the shared sparsifying transform, Bob as the client generates a sparse representation from his query data and subsequently sends it to the server.

Server Step 2 as a matching test for the client: The server searches to find the most similar sparsecode set, and subsequently replies to the client’s request.
In fact, Alice has a secret message while she decides to send it to Bob in the presence of the passive potential or/and active actual Eve(s).
IiC Main problem
A sparse dictionary learning problem can be mathematically formulated as [1, 2, 3, 4, 5, 6]
(1) 
with regard to the thresholds and , where denotes the observed data, represents the unknown dictionary, stands for the original data set w.r.t. the data instance , refers to the sparse representation coefficient set, is a distribution over appropriate perturbations [6], while we have the pretrained models and .
Definition 1^{3}^{3}3See e.g. [19, 20].: secret key.
with the finite range informationtheoretically represents an secret key for Alice and Bob, achievable through the public communication , if there exist two functions and such that for any the following three conditions hold: (i) ; (ii) ; and (iii). These three conditions respectively ensure that (i) Alice and Bob generate the same secret key with high probability as much as possible; the generated secret key is efficiently kept hide from any user except for Bob who gets access to the public communication
; and (iii) the secret key is subuniformly distributed.
This definition is based fully upon a passive Eve which can be extended to an active scenario although we in the next parts, use the active one.
Iii Main results
In this section, our main results are theoretically provided in details.
Iiia Time evolution of the sparsity
Proposition 1
The time evolution of the sparsity values, that is, as DoF relating to the transition jumps for the ratedistortion based sparse coding optimisation problem expressed in Eqn. (1) can be mathematically attainable for the distortion term while for the instant .
Proof: See Appendix A.
IiiB Relaxation of the ratedistorion problem
Proposition 2
While the degrees of freedom of interest so far is , following the previous result, i.e., Proposition 1 and its relative proof given in Appendix A, the main problem over the distortion term is hard. However, this problem can be theoretically relaxable through a new definition over the given sideinformation as a new distortion constraint .
Proof: See Appendix B.
IiiC Positive definity
Proposition 3
DoF, if they are nonzero, they are conditionally positive definite.
Proof: See Appendix C.
IiiD Extension to the eavesdropping scenraio
Assume^{4}^{4}4See e.g. [21, 22]. that we have two users, without loss of generality one of which is Bob as the legitimate user, and another one is an eavesdropper call Eve who can get access to the information in relation to the private message arose between Alice and Bob.
Proposition 4
For the th instant, call the sparsity patterns and respectively for Bob and Eve, that is, in relation to the channel between Alice and Bob, and the one between Alice and Eve, respectively. For the th instant and th subchannel, define^{5}^{5}5See e.g. [21, 22]. . The Problem expressed in Eqn. (6) can be rewritten as the Problem expressed in Eqn. (7), or the Problem expressed in Eqn. (8) or even the Problem expressed in Eqn. (9).
Proof: See Appendix D.
Iv Numerical results
We have done our simulations w.r.t. the Bernoullidistributed datasets using GNU Octave of version
on Ubuntu .Fig. 2 shows the optimum value earned from the Problem given in Eqn. (6) while and are divided by the one when , i.e., the case in which we do not constrain our optimisation problem w.r.t. the relative constraints. As totally, obvious, the more constraints we intelligently and purposefully provide, the significantly more favourable response we can undoubtedly guarantee with regard to our proposed scheme.
Fig. 3 shows the instantaneous keyrate earned from the Problem given in Eqn. (9) while and are divided by the one when , i.e., the case in which we do not constrain our optimisation problem w.r.t. the relative constraints. The same as the previous trial, the more constraints we smartly provide, the considerably more adequate response we consequently assure in relation to our proposed scheme.
V conclusion
We have theoretically shown a novel solution to an hard ratedistortion based sparse dictionary learning in the context of a novel method of relaxation. We theoretically showed the accessibility of DoF in which er are interested via a Langevin equation. In fact, the relative time evolution of DoF, i.e., the transition jumps for a relaxation over the relative optimisation problem was calculated in the context of a tractable fashion. The aforementioned relaxation was proven with the aid of the Graphon principle w.r.t. a stochastic Chordal SchrammLoewner evolution etc. Our proposed method and solutions were extended to the eavesdropping case. The effectiveness of our proposed framework was ultimately shown by simulations.
Appendix A Proof of Proposition 1
The problem written in Eqn. (1) can be theoretically reexpressed as
(2) 
while for which holds, or correspondingly in the context of an instantaneous^{6}^{6}6See e.g. [23, 24] to understand what an instantaneous expectedvalue it means. optimisation problem
(3) 
Now, as we are interested in theoretically exploring the sparsity, as DoF in relation to the transition jumps, one can apply the Langevin equation according to which the time evolution of the aforementioned DoF can be accessible. Thus, we have
(4) 
while is an arbitrary parameter, and the minus sign principally shows a backward orientation in the solution to the problem, as well as the theoretical fact that is a randomwalk.
The proof is now completed.
Appendix B Proof of Proposition 2
The proof is provided here in terms of the following multistep solution.
Step 1. It is strongly conventional [25, 26] to define a constraint over a distortion between the sideinformation and the sparse data set. Therefore, we do this here as well according to which our optimisation problem is conveniently rewritten as
(5) 
w.r.t. the arbitary threshold .
Step 2. One can here say that DoF cited above can be in terms of the possibly available ways via which one can calculate a specific amount of information. Accordingly, the possibly available pieces of a specific amount of information calculable such as the relative sideinformation set can be DoF intersted here to be calculated.
Step 3. Now, assume that the possibly available pieces of the specific amount of information calculable talked above are in the context of probabilistic energy^{7}^{7}7In order to see the relevance between energy and information see e.g. [27, 28, 29]. quanta. Subsequently, we assume a given Ball in which the number of energy quanta should require
while is in relation to , and we have a probability density function (pdf) while , and stand for the energy, momentum and time, respectively. Subsequently, we assign
while is the second derivative in the polar axis. Thus, defining
the relative energy quanta numbers is calculated as
(6) 
(7) 
(8) 
(9) 
Step 4. Now, theoretically imagine a game and the relative users as follows. The Reward is each of the possibly available pieces of the specific amount of information calculable in the previous step, for each relative DoF as the users.
Remark 1. In our game, the population physically has a positive growth rate. This is because of the fact that the principle of DiffusionandReaction hold here, that is, a diffusive flow exists from the side of a higher concentration to a less one, and reciprocally, a reactive movement exists from the less concentration towards the higher one.
Step 5. Now, let us mathematically define an initial condition for the Langevin based constraint. We have . Now, call the PicardLindelof Theorem^{8}^{8}8See e.g. [30] to see what it is. which can be of a purely useful nature to say that: the Langevin equation
w.r.t. the initial condition can have a unique solution w.r.t. a surely existing over the zone ; while is uniformly Lipschitz continuous in and continuous in . Thus, we indeed branch and localise the optimisation problem into its localities, without loss of generality and optimality. This is because of the hard structure of our main optimisation problem since that is not convex, so, the strong duality does not theoretically hold [31].
Step 6. Recall Remark 1 for the fact that the population size of the relative users has a growth over the time zone, according to which we can theoretically consider a Graphon^{9}^{9}9See e.g. [6]. in between. Indeed, we can prove that there exists a duality between the main optimisation problem and its localised version created in Step 5. Now, w.r.t. the fact that there would exist a stochastic Chordal^{10}^{10}10Which maps a boundary point to another one, which is in contrast to the Radial SchrammLoewner evolution which maps a boundary point to an interior one: See e.g. [32, 33] to see what they are. SchrammLoewner evolution^{11}^{11}11See e.g. [32, 33] to see what it is. which shows a Brownian motion over the surface of the relative evolving Riemannianmanifold relating to the graph as
w.r.t. the arbitary function and the randomwalk . On other hand, since a graphon as a density measure is the number of times occurs as a subgraph of the graph , and w.r.t. the fact that two graphs, do they have the final approximation in terms of a same graphon, they actualise the same phenomenon [34], and regarding the fact that the positive growth of the population size literally defined above highly requires an adjacent matrix for the evolving Riemannianmanifold and the graph in between, a proof for a the duality we are looking for can be proven. Now, the duality basically holds iff and only iff and conclude a common phenomenon, while fundamentally is a stochastic SchrammLoewner evolutioned version of in the nearest physical vicinity of the Reimannian submanifold, relating to the surface of the relative evolving Riemannianmanifold, that is
while is a sufficiently small nonzero value and and respectively show the final realisation of the relative graphs and . This is equivalent to^{12}^{12}12This kind of optimisation problem can be solved by the alternating direction method of multipliers: See e.g. [27, 28, 29].
Towards such end, we reexpress our optimisation problem as Eqn. (6) on the top of the next page in which the term indicates the stochastic SchrammLoewner evolution expressed above and is an arbitary threshold to constraint the amount of the SchrammLoewner evolution. Now, let us continue by the following definition.
Definition 2^{13}^{13}13See e.g. [35].: The order binomial extension. Consider the mapping such that holds and a
which is a discrete random variable that takes values on the nonzero nonnegative integers
. The mapping such that holds is theoretically called the order binomial extension of if .In the subsequence of Definition 2, one can reconsider the optimisation
as
while basically is the probability mass function (pmf) of . It is also importantly interesting to be noticed that the term comes principally from the convolutional nature of the Graph associated fundamentally with since the terms and may be technically of a Poissonrandomlydistributed nature. Meanwhile, we may interchangeably use with a slight abuse of notation here, without loss of generality.
Now, consider the graph and the subgraphs and with respectively random vertex sets , and . Let , and be the probability density distributions respectively on , and whose marginal on are equal to . Now the later optimisation can be rewritten as
where holds, w.r.t. the nonzero nonnegative threshold as well as arbitary functions and , while the first two additional constraints are thoroughly justified according to the Korner’s graph entropy^{14}^{14}14[36, 37]. something that can be stongly followed up via Riemannian gradients in Riemannian geometry as described below. Indeed, the first twoconstraints have come from the fact that is equivalent to
The strongly theoretical logic behind of our interpretation and consideration over the evolution expressed above can be physically justified as follows as well, according to the Riemannian geometry and Riemannian optimisation principles^{15}^{15}15 See e.g. [38, 39, 40].. Let us initially define the Riemannian gradients^{16}^{16}16 See e.g. [38, 39, 40, 41, 42]. in relation to the submanifolds as the orthogonal projection of the distortion function
(10) 
onto the associated tangent space as
which can be widely rewritten in terms of the Riemannian logarithmic map operator
while and theoretically are the eigenvalue matrices associated with the graphs and , respectively. Now, defining translation functions and respectively for the realparts of and , and rotation functions and for the relative imaginaryparts, while the following conditions satisify
while principally indicates the average value and and are Brownian motions. Now, our relaxation we are focusing on is equivalent ot the following optimisation
Lemma 1
For the equation derived above, the necessary and sufficient condition to be relaxable fundamentally is the fact that there should exist and such that the following satisfies
Proof: The proof is omitted for brevity.
Step 7. Finally, the following lemma can help us in concluding our interpretation.
Lemma 2
The constraint
is interpreted according to the Bochner’s theorem.
Proof: The Bochner’s theorem^{17}^{17}17see e.g. [43] to see what it is. says that for a locally compact abelian group^{18}^{18}18A group while each pair of group members are interchangible. , with dual group , there exists a unique probability measure such that
holds. This completes the proof of Lemma 1 here.
The proof is now completed.
Appendix C Proof of Proposition 3
The HerglotzRiesz representation theorem for harmonic functions^{19}^{19}19see e.g. [44] to see what it is. can be useful here for us saying that DoF we are interested in, if they are nonzero, have positive realparts iff and only iff there is a probability measure on the unit circle such that
holds, while
since it is proven that is harmonic, that is
according to the Newton’s second law of motion, roughly speaking.
The minimisation defined above can be rewritten as
for the case where is holomorphic.
Remark 3. The term can be reexpressed as
while . In this term, the term shows the story in relation to the tangent space expressed in Step 6 in Appendix B.
The proof is now completed.
Appendix D Proof of Proposition 4
The achievable secretkey rate per dimension is^{20}^{20}20see e.g. [21, 22].
while holds, and is the instantaneous signaltonoise (SNR), as well as the fact that
and
hold^{21}^{21}21see e.g. [21, 22]., while is the observed data by Eve.
Now, one can assert that the Problem expressed in Eqn. (6) can be rewritten as the Problem expressed in Eqn. (7) or the Problem expressed in Eqn. (8), while for the th instant and th subchannel, we have
and we principally define the following probabilistic constraint
which can be in parallel with
w.r.t. the arbitrary thresholds , , and , roughly speaking.
The two probabilistic constraints theoretically introduced here stand respectively for the following facts: (i) the upperbound of the probability value should be constrained through a threshold; (ii) and the lowerbound of the instantaneous SNR should also be constrained through a threshold as well.
Instead of the two constraints given above, one can constrain the secretkey rate as
w.r.t. the arbitrary thresholds and .
Remark 4. It is absolutely interesting to note that although perturbations are considered in the literature mainly due to attacks, we do not ignore the term in Eqns. (789) in order to preserve the optimality and generality of the problem.
Remark 5. The three probabilistic constraints newly declared above are relaxable according to the principle of the ConcentrationofMeasure something that theoretically proves an exponential bound for the two aforementioned constraints^{22}^{22}22See e.g. [27, 28, 29] for further discussion..
The proof is now completed.
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