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Capacity of Remote Classification Over Wireless Channels

Wireless connectivity creates a computing paradigm that merges communication and inference. A basic operation in this paradigm is the one where a device offloads classification tasks to the edge servers. We term this remote classification, with a potential to enable intelligent applications. Remote classification is challenged by the finite and variable data rate of the wireless channel, which affects the capability to transfer high-dimensional features and thus limits the classification resolution. We introduce a set of metrics under the name of classification capacity that are defined as the maximum number of classes that can be discerned over a given communication channel while meeting a target classification error probability. The objective is to choose a subset of classes from a library that offers satisfactory performance over a given channel. We treat two cases of subset selection. First, a device can select the subset by pruning the class library until arriving at a subset that meets the targeted error probability while maximizing the classification capacity. Adopting a subspace data model, we prove the equivalence of classification capacity maximization to Grassmannian packing. The results show that the classification capacity grows exponentially with the instantaneous communication rate, and super-exponentially with the dimensions of each data cluster. This also holds for ergodic and outage capacities with fading if the instantaneous rate is replaced with an average rate and a fixed rate, respectively. In the second case, a device has a preference of class subset for every communication rate, which is modeled as an instance of uniformly sampling the library. Without class selection, the classification capacity and its ergodic and outage counterparts are proved to scale linearly with their corresponding communication rates instead of the exponential growth in the last case.


I. INTRODUCTION
There is an emerging trend of deploying various Artificial Intelligent (AI) algorithms at the edge, away from the central cloud, to provide a context-aware and low-latency platform for supporting a wide range of applications such as Internet search (e.g., Google Lens), digital payment (e.g., Alipay's Smile to Pay), and inter-connected vehicles in 5G [1]. The ubiquitous wireless connectivity results in a new paradigm merging communication and inference, called edge inference, referring to the broad set of techniques for deploying trained AI models at edge servers to remotely execute inference tasks posed by mobile users, such as object recognition or speech interpretation.
A large class of edge inference services can be reduced to the model in which a mobile user wirelessly uploads a multimedia data sample (photo, video or speech clip) over a wireless link, the edge server recognizes an object embedded in the sample and feeds back the object label.
We term this operation remote classification and it is the main theme of this work. A large-scale remote classifier in the edge/central cloud is capable of rendering many object classes, around 700 for Google Cloud and 200 for Tencent Cloud. Maximizing the correctness of classification requires a user to upload high-dimensional features (or large-size raw data). However, this is challenged, on the one hand, by the variability of the wireless link due to fading and interference and, on the other hand, by the stringent latency requirements in real time and/or high-mobility applications. To address this issue, an existing remote classification service achieves the required versatility by deploying a system of classifiers with diversified capacities, which are switched according to the application requirements, input data quality, or dimensionality of input feature vectors. Motivated by this, we study the capacity of remote classification as a function of the communication rate offered by the wireless link.

A. Classification, Channel Coding, and Source Coding
The essence of the remote classification problem can be better understood by relating it to two classic problems in information theory: source coding and channel coding. In source coding (or compression), a transmitter represents source information using codewords that can be sent over a limited-rate channel and enable the receiver to accurately reconstruct the information [3].
In channel coding, the transmitter selects a set of codewords to which the messages are mapped and the receiver should be capable of differentiating the codewords even in the presence of channel noise, thereby decoding transmitted messages [4]. Source coding can be seen as a process of remote estimation, while channel coding as a process of remote classification in which codewords are subject to design. Remote classification can be related to coding based on the following interpretation. We can view class as "codewords" chosen by the nature and the objects as noisy instances of the classes [5]. Then the transmitter sends a description (features) of a noisy instance over a limited-rate channel, such that the receiver is able to "decode" (infer) the "codeword" (the covert class or the label of the instance). Therefore, the name of "remote classification" as used in this paper, refers to a particular remote classification process in which the classes ("codewords") are not subject to design.
Despite the similarity, there exist several fundamental differences between remote classification and source/channel coding. First, the "codewords" (classes) in the former are chosen by the nature and not subject to design as in the latter. As a result, a typical multimedia classifier cannot be derived theoretically. Instead, it is usually computed using a supervised machine learning technique, which includes choosing a suitable model [e.g., support vector machine (SVM) or convolutional neural network (CNN)] and training the model using a large labeled dataset [5].
Second, in source/channel coding, it is the transmitter that has the ground-truth information while the receiver gets an imperfect version of this information. In contrast, in remote classification, the receiver is the one responsible for inferring the ground-truth information (in the form of a label); the transmitter does not have the information and acquires it via a feedback channel. Finally, the general problem of multimedia classification can have different mathematical characteristics from those of coding such as data spaces (e.g., a feature space versus a Galois field).
Despite the differences, relating remote classification to source/channel coding creates the possibility of exploiting analytical tools from the rich literature on the latter to study the former.
An early work in this direction is [6] where the rate of a stand-alone classifier is found to be mathematically equivalent to the capacity of a MIMO channel with space-time modulation. In this work, we adopt a similar approach to investigate the performance of a different system of remote classification featuring a pair of separated classifier and data source that are connected using a wireless channel.

B. Edge Computing and Inference
Remote classification and edge inference at large are services supported on the edge computing architecture [1]. The current work shares the same spirit as that on computation offloading, a main theme of edge computing research, where mobile devices use unreliable wireless links July 31, 2020 DRAFT to offload computation to edge servers. In the current work, this computation is exemplified by classification. Edge computing augments the capabilities of mobile devices while preserving their energy efficiency [7]. To reduce the devices' energy consumption, a key approach for energy efficient computation offloading is to jointly optimize radio resource allocation to multiple users and their offloaded computation loads [8]- [10]. Stochastic optimization tools, such as Lyapunov optimization, are applied to adapting offloading decisions [10] and servers' CPU frequencies [11] to the dynamics in computation tasks and channels in order to reduce both latency and power consumption. More complex techniques for accelerating offloaded computation include replicated computation at multiple servers [12], adding the new dimension of caching to the joint communication-and-computation control [13], and scheduling of computation tasks [14]. Without considering a specific application, the prior work is based on generic computation models, such that the load is measured by the number of bits and the speed by the number of bits computed per second.
Attempts on materializing the vision of edge AI has led to the emergence of edge learning (see, e.g., [26], [27], for an overview) and edge inference, which is the theme of this work.
Research in edge inference has resulted in several interesting design approaches. Building on the mentioned idea of replicated computation in [12], it is proposed in [15] that the association between servers (base stations) and devices can be optimized together with beamforming to reduce the total energy consumption of the devices. Several research groups have developed techniques to implement device-edge cooperative inference, where a learning task is partitioned and executed partially on device and partially offloaded to the servers [16]- [18]. To address the issue of limited computation capacity of a device, a CNN model can be pruned before partitioning, and the idea can be implemented using the techniques in [16]. There also exist techniques for channel adaptive model partitioning and coding [17]. Furthermore, the model partitioning can be adjusted according to the allocated bandwidth and the requirements on latency and inference accuracy, which is the approach advocated in [18]. In addition, data compression for communication-efficient edge inference has also been investigated. For example, a relevant architecture is proposed in [19] where a deep neural network (DNN) encoder is deployed at a transmitter to compress raw data and the compressed data is decoded by the server using a DNN encoder before feeding the output into another DNN model for inference. In view of prior works, they are focused on technique design and rely on experiments for performance evaluation. There exist few results on the fundamental limits of edge inference systems under the constraint of July 31, 2020 DRAFT wireless channels connecting severs and devices, which motivates the current work.

C. Contributions and Organization
The objective of this work is to make the first attempt on quantifying the performance of a remote classification system under a communication channel constraint, referring to the finite and time-varying rate of a wireless communication channel. To this end, we consider a system in which a mobile device sends a feature vector over a wireless channel to an edge server, which performs classification and sends the result to the mobile device. The server supports classification of an arbitrary subset of a class library based on a mainstream architecture of large-scale classification (see e.g., [20]). On the one hand, even if the communication rate is sufficiently large for transmitting all features of each sample, classification errors can still occur as an inherent effect of data noise, which is caused by the natural factors in sensing (e.g., pose, perspective, lighting, and background). On the other hand, as the rate varies and so does the received number of features per sample, if the classification error probability should be constrained, the maximum number of object classes that are chosen to be discerned by the remote classifier has to be adapted to the rate in a similar way as the maximum constellation order of adaptive modulation. This gives rise to a performance metric called -classification capacity defined as the maximum number of classes that can be discriminated under the channel constraint and for a given target classification error probability 1 . Furthermore, two derivative metrics, called ergodic and outage classification capacities, are defined to account for the effect of fading, which correspond to adaptive and fixed coding rates, respectively. Using these metrics, the system performance is analyzed for two cases.
• Class-selection case: The user has broad interests covering the whole library (e.g., augmented reality). Given a communication rate, the user selects a subset of classes for classification with the aim of maximizing the classification capacity while meeting the target error probability.  The main contributions of the work are summarized as follows.
• Classification capacity with class selection: Consider the mentioned class-selection case. Organization: The remainder of the paper is organized as follows. The models and performance metrics are introduced in Section II. Section III presents the analysis on classification capacities with class selection while that for the random-class case is investigated in Section IV. The derived results are further extended to fast fading channels in Section V. Section VI provides the experimental results, followed by concluding remarks in Section VII.

II. MODELS AND METRICS
Consider the remote classification system in Fig. 1, where an edge device transmits feature vectors, extracted from data samples, to an edge server for classification using a trained model and receives from the server the inferred labels. The specific models and performance metrics are described as follows.

A. Classification Model
As in [6], we consider the classic statistical problem of classifying linear subspaces. The statistical data model and ML classifier are described as follows.
1) Statistical data model: Consider a clustered dataset comprising L separable classes, where the i-th class centroid is represented by a unitary matrix U i ∈ R N ×K with N ≥ K and U T i U i = I K . An arbitrary data sample, denoted asx, that belongs to the i-th class is modeled as [6] x = ΦU i s +w, where the unitary matrix Φ represents the discriminant subspace embedded in the raw-data space, s results from the projection of the data sample into the class subspace U i , andw accounts for both the error in fitting the dataset distribution to the subspace model as well as the mentioned data noise. Note thatw is the inherent cause of classification errors even in the absence of channel constraint. The random vector s ∈ R K is assumed to consist of independent and identically distributed (i.i.d.) N (0, σ 2 s ) elements. To compress the sample, a feature vector, denoted as x, is extracted fromx by projecting it onto the discriminant subspace: where w = Φ Tw comprises i.i.d. N (0, σ 2 w ) elements and referred hereafter simply as data noise. The subspace Φ is assumed to be known to the server for calibrating the needed classifier; when Φ is determined by the sever, the operation is known in the literature as feature selection. Based on (2), the data model can be parameterized by the subspace set U L = {U }.

Definition 1. (Data SNR).
The signal-to-noise ratio (SNR) of the dataset is defined as the ratio between the variance of each cluster and that of data noise: where we set σ 2 w = 1 without loss of generality.
2) Maximum-likelihood remote classifier: Conditioned on U i , the probability density function (PDF) of x is given as Given the knowledge of {U i } L i=1 , the classifier estimates the class of a reliably received feature vector x, say U i , (or equivalently the label i) by maximizing the above PDF: which is the well-known ML classifier.

Remark 1. (Geometric Interpretation).
The operation U i U T i x projects the feature vector x onto the subspace, span{U i }. It gives the geometric interpretation that the ML classifier essentially aims at identifying the subspace forming the smallest angle with (or equivalently having the smallest subspace distance to) the feature vector x.

B. Communication Model
Time is divided into slots, each of which has the duration of T seconds. Each feature is quantized into a sufficiently large number of bits, denoted as Q, such that distortion is negligible.
The channel code is designed such that each quantized feature vector is encoded into a single codeword transmitted using one slot. The variation of the channel with bandwidth B is assumed to be slow w.r.t. the slot duration such that the channel remains constant within each slot but varies over slots. The extension to the scenario of fast channel variation is presented in Section V. Let R denote the communication rate (bit/s) of the channel. Both the cases of channel adaptive and fixed coding rates are considered as discussed in the sequel. As an example, given the transmit SNR (denoted as ρ) and without channel state information at the transmitter (CSIT), the rate where h denotes the channel gain. As another example, the rate for a multiple-input-multiple-output (MIMO) channel is R = B log 2 det I + ρ Nt HH H where H denotes the channel matrix and N t the number of transmit antennas. The finite communication rate introduces a constraint on the feature dimension

C. Performance Metrics
To facilitate defining the performance metrics, the notion of (object) class library is first formalized. In practical remote-classification, the server supports a large library of classes and can generate an active classifier for a user based on the chosen subset of classes (see e.g., [20]).
The class library is represented by F = {F 1 , F 2 , ..., F M } where each element is a subspace matrix representing an available class. The L-class subset chosen by a user is specified by the subspace set U L with U L ⊂ F, which determines the dataset distribution.
1) Classification error probability: Labels inferred by the remote classifier can be erroneous as an inherent effect of data noise even though the channel is reliable and even if its rate is sufficiently large to transfer all features. A classification error is declared if the inferred label is different from the ground truth. Conditioned on the data distribution specified by U L and the communication rate R, the classification error probability, denoted as P e , can be written as July 31, 2020 DRAFT where L denotes the classifier function mapping the input feature vector to the inferred label and y is the ground-truth label. Note the above definition assumes that the prior probability that the object x belongs to one of the L classes is uniform, as in [6] and R determines the length of x as described in the sequel. The future extension to the case with non-uniform probabilities requires modifying the classifier model by adding prior-dependent weights to the likelihoods of different labels.
2) -classification capacity: Recall that the metric, denoted as C, is defined as the maximum number of classes that can be discriminated given an instantaneous communication rate, R, such that the classification error probability, P e , is no larger than a given threshold ∈ (0, 1).
Conditioned on R and U L , the error probability can be written as the function P e (R, U L ). Using the notation, the -classification capacity for the class-selection case can be defined as: The counterpart for the random-class case is defined as where the expectation is over the distribution of the classes, U L , given L.

3) Ergodic classification capacity:
Consider the case where the device has CSIT and adapts the number of features per sample as well as coding rate to the channel state. Then we can define the ergodic classification capacity as: where the expectations are over the distribution of communication rate R, C sel (R) is defined in (7) and C rnd (R) in (8).

4) Outage classification capacity:
A different communication model is adopted where either the CSIT is unavailable or some form of channel inversion is used such that the channel cannot be inverted when its gain is below a given threshold. As a result, the device fixes the number of features per sample and coding rate, resulting in a required communication rate, r, for successful decoding of a received feature vector. Then a channel outage event is one that the channel capacity falls below a given threshold r, yielding the outage probability defined as Under an outage constraint, P out ≤ δ, and a fixed transmit SNR, there exists a maximum rate of r. Then the outage classification capacity is defined as where C sel (r) and C rnd (r) are defined in (7) and (8), respectively.

Remark 2. (Effective Classification Error Probability).
For remote classification, when the channel is in outage, the server receives zero features for a transmitted sample but may make a random guess on the sample's label with the error probability of L−1 L . If this is the case, the effective classification error probability is slightly larger than its constraint and should be given

III. CLASSIFICATION CAPACITY WITH CLASS SELECTION
In this section, the -classification capacity and its ergodic and outage counterparts for the class-selection case are analyzed.

A. Classification Error Probability
To facilitate the derivation of classification capacities under a constraint on the classification error probability, we first analyze the probability as follows.
1) Pairwise classification error probability: Consider the classification of two specific classes, namely U i and U j . The error probability of binary classification based on a similar data distribution model as the current one was studied in [24] in the context of space-time demodulation.
Let "i → j" denote the event that a sample of class i is assigned label j by the classifier.
Then the pairwise classification error probability (PCEP) can be defined as P (i → j) = Pr (L(x) = j |y = i, U L , R). A main result from [24] is given below. [24]). The probability is given as

Lemma 1. (Exact Pairwise Classification Error Probability
where θ (i,j) k denotes the k-th principal angle between U i and U j , and a k = 1 4 + Note that the effect of the number of features per sample, N , (or the proportional communication rate, R) is not reflected in the above result given a fixed distance between U i and U j , measured by {cos 2 θ where g(σ 2 s ) = is a monotonically increasing function of the data SNR σ 2 s and subspace distance between the two classes U i and U j .
Remark 3. (Effects of Data SNR and Class Distance). On the one hand, increasing the data SNR causes data clusters to shrink, improving their discernibility. For this reason, it is observed that both bounds on the PCEP in the above lemma decrease as the data SNR grows. On the other hand, the subspace distance between two classes determines their differentiability. Consequently, increasing the distance reduces the bounds on the PCEP. The improvement is known as the discrimination gain in the literature.
where (a) is due to the union bound and (b) due to the symmetry P (i → j) = P (j → i). Define d min = min (i,j) d i,j , the above bound can be further relaxed to give the upper bound in Lemma 3 July 31, 2020 DRAFT in the sequel.
Next, P e can be lower bounded as follows. Define W i,j * as an event that the ground-truth min . Then it follows from (6) that one lower bound of the classification error probability can be calculated as yielding the lower bound in Lemma 3.

Lemma 3. (Classification Error Probability).
Given the class subspace set U L , the classification error probability can be bounded as

Remark 4. (Effect of Number of Classes).
Apart from the effects of data SNR and class distance discussed earlier, one can further observe/infer from the above bounds that increasing the number of classes, L, makes the classification error probability grow. This is because that packing more classes into a fixed feature space reduces inter-class distances and thereby compromise their differentiability.

B. -Classification Capacity
The key step in deriving the -classification capacity is to establish the equivalence between the classification capacity maximization via class selection and the Grassmannian packing problem.
To facilitate the analysis, we consider the scenario where the large-scale remote classifier at the server can support flexible classification as stated in the following assumption. In practice, large-scale classification realizes flexible classification using a hierarchical architecture comprising a large number of component classifiers [20], [21].  (7) and Assumption 1, the subspace set U * that represents the class selection for capacity maximization can be found by solving the following optimization problem: where G = G(N, K), C(U) = C(R, U) and P e (U) = P e (R, U) with N , K, and R in this subsection and omitted to simplify notation. Substituting the upper bound on P e in Lemma 3 into (P1), the problem can be recast as . The solution of (P2) lower bounds the maximum capacity from solving (P1) and the approximation is accurate when the error probability is small. An intuitive interpretation of Problem (P2) is to pack as many balls as possible (maximizing L), each centered at an element of U L and with the radius d min 2 , into the space G, giving the name Grassmannian packing [22]. A standard approach for solving this class of mathematical problems is to convert them into equivalent problems of maximizing the minimum separation distance among L balls [22]: Let d * min (L) denote the result from solving the above problem, called minimum class separation from packing. Then, L is increased to reach the maximum value under the constraint d * min (L) ≥ β L , thereby solving the original Problem (P1).
Though typically Grassmannian packing problems are intractable and usually solved numerically, there exists a rich literature on bounding the resultant minimum distance d * min (L). (see e.g., [22]). The following particular result is from [23].

Lemma 4. (Packing Bounds).
For large feature-space dimensions N , the minimum class sepa-ration distance from Grassmannian packing can be bounded as 2) Packing bounds on -classification capacity: Using Lemmas 3 and 4, the -classification capacity defined in (7) can be bounded as C lb ≤ C sel (R) ≤ C ub with C lb = L : L 2 C ub = L : 1 3 where and (19) Note that · is omitted as d 2 lb → d 2 lb for large N and K, which is a typical case in multimedia classification. Solving the two equations (18) and (19) and substituting N = βR yield the following theorem.

Theorem 1. ( -Classification Capacity with Class Selection). Consider the class-selection case.
For a high communication rate, the capacity can be asymptotically bounded as: where c σ 2 s = log 2 log 2 (1 + g(σ 2 s )) and c = log 2 log 2 1+g(σ 2 s ) 2 . In particular, as R, K → ∞, the capacity scales as lim R,K→∞ Proof: See Appendix B.
Remark 5. (Mathematical Intuition for Capacity Scaling Laws). One can observe from the above theorem that the -classification capacity increases exponentially as the instantaneous communication rate R grows. The underpinning mathematical reason is that the volume of the Grassmann manifold containing the dataset classes is an exponential function of its dimensions N , which is proportional to R. Consequently, increasing R allows an exponentially growing number of "balls" (classes) to be packed into the manifold. One the other hand, the capacity scales super-exponentially with the dimensions of each data cluster (or each class), namely K. Note that increasing K improves the inter-class differentiability. One can infer from (18) July 31, 2020 DRAFT and (19) that with the classification error probability fixed, the allowed number of "balls" (L) grows exponentially as the minimum pairwise distance of the "balls" (classes), d * min , increases. Furthermore, d * min is a super-linear function of K as one can further observe from the definitions of d 2 ub and d 2 lb after (19). Combining the two relations gives the super-exponential capacity scaling w.r.t. K. Remark 6. (Effects of QoE Requirement and Data/Transmit SNR). The dependence of the -classification capacity on the allowed maximum classification error probability (or QoE requirement), the data SNR σ 2 s , and the transmit SNR ρ can be interpreted geometrically in terms of Grassmannian packing. Increasing , σ 2 s and ρ allows "balls" (classes) to get closer, shrinks "ball radiuses" (the variance of each data cluster), and increasing the Grassmannian volume (communication rate), respectively. They all contribute to packing more "balls" (larger capacity) though in different ways.

C. Ergodic and Outage Classification Capacities
Given a distribution function of the communication rate R, it is straightforward to use the results in Theorem 1 to analyze the ergodic and outage classification capacities based on their definitions in (9) and (11). In this section, we consider a Rayleigh fading channel and perform such analysis to provide concrete insight into the effect of channel fading on the performance of remote classification.

Proposition 1. (Ergodic Classification Capacity for Rayleigh Fading). Consider the class-selection
case. The ergodic classification capacity defined in (9) can be bounded as where γ lb = βB 2 (K log 2 K + Kc σs − Kc ) and γ ub = βB with c σs and c defined in Theorem 1. In particular, for largeR and K, the capacity scales as where β = T Q .
Proof: See Appendix C.

Remark 7. (Fading Does Not Affect Capacity Scaling).
The key observation from the above proposition is that both the scaling laws of the ergodic classification capacity are the same as those for -classification capacity in Theorem 1 except for replacing the instantaneous rate R with its ergodic counterpartR. The remark also applies to outage classification capacity analyzed in the sequel if the communication rate is modified as the maximum rate under an outage constraint.
2) Outage classification capacity: To begin with, the maximum communication rate, denoted as R δ , can be obtained from the active outage constraint P out = Pr(R ≤ R δ ) ≤ δ and the exponential distribution of the channel gain as It is worth mentioning that R δ is a monotonically increasing function of the outage probability δ.
Moreover, note that the corresponding number of transmitted features per sample is now given as N = βR δ . The outage classification capacity is equal to the -classification capacity by replacing R with R δ in (24), yielding the following proposition.

Proposition 2. (Outage Classification Capacity for Rayleigh Fading) Consider the class-selection
case. The outage classification capacity defined in (11) can be bounded as with ρ 1, γ lb and γ ub defined in Proposition 1. In particular, as R δ , K → ∞, the capacity scales as Proof: See Appendix D.

IV. CLASSIFICATION CAPACITY WITH RANDOM CLASSES
In this section, the -classification capacity and its ergodic and outage counterparts are analyzed for the random-class case and compared with their counterparts in the class-selection case.

A. Expected Classification Error Probability
The expected classification error probability is analyzed in this subsection for a dataset with i.i.d. isotropic classes, {U }, on the Grassmannian G(N, K).

1) Distribution of class separation:
Let θ max denote the maximum principal angle between a pair of classes, U i and U j .
The squared chordal distance between U i and U j is defined as d 2 Proof: See Appendix E.
2) Expected classification error probability: Consider the ML classification of two random classes. Using Lemmas 2 and 6, the expected PCEP can be bounded as follows.

Lemma 7. (Upper Bound on Expected PCEP). For a pair of independent and isotropic random
classes U i and U j , the expected PCEP can be upper-bounded as Proof: See Appendix F.
By applying the union bound and using Lemma 7, we obtain the following lemma.

B. -Classification Capacity
The -classification capacity defined in (8) can be obtained by solving: By modifying the constraint using (30), the capacity can be lower-bounded as follows.

Theorem 2. ( -Classification Capacity with Random Classes). For a large communication rate,
the -classification capacity for the random-class case can be asymptotically bounded as:

C. Ergodic and Outage Classification Capacities
The linear scaling of the -classification capacity w.r.t. to the communication rate R makes it straightforward to extend the result to ergodic and outage classification capacities by modifying R accordingly, giving the following proposition.
Proposition 3. (Ergodic and Outage Classification Capacities with Random Classes). The ergodic and outage classification capacities for the random-class case can be bounded as as N x . This also holds in the random-class case for a different reason that random erasures of some dimensions of the feature space does not change the isotropic distribution in the resultant space. The random variable N x is determined by the number of successfully received packets, X, that follows the binomial distribution: Given the average number of successfully received packets, (1 − η)S, fixed, for large S, the distribution can be approximated as Poisson: 1) Class-selection case: Combining the approximate distribution function, N x = nN S and Theorem 1, the ergodic classification capacity is derived as 2) Random-class case: The ergodic classification capacity in this case can be easily modified from its slow-fading counterpart in Proposition 3 by redefining the ergodic communication rate for the current case: whereR = (1−η)N β . As before, the effect of fast fading is to scale the ergodic classification capacity by the packet-success probability (1 − η).

A. Experimental Settings
Two sets of experimental results are obtained based on the statistical data model used in the preceding analysis and a real dataset, respectively. Their corresponding experiment settings are as follows. For all experiments, fading is modeled as Rayleigh, the transmit SNR is set as 15 dB and channel bandwidth as 50 KHz.
• Statistical data model: The selected Grassmannian packing datasets were generated by Conway and Sloane [28]. The maximum classification error probability is 0.03 and 0.19 for the class-selection and random-class cases, respectively, and the maximum (channel) outage probability is 0.3. The data SNR is set as 15 dB.
• Real dataset: The well known MNIST dataset is used that comprises images of handwritten numbers. For inference, the popular neural network model, multi-layer perception (MLP), is adopted as the classifier and trained using the training dataset of MNIST. The maximum classification error probability is set as 0.02.  Fig. 2(c) and (d).

C. Classification Capacities with Real Dataset
Available class subsets are generated by different combinations of classes in the MNIST dataset. The example of three 3-class subsets is illustrated in Fig. 3.
The -classification capacities for the cases of class selection and random classes are compared in Fig. 4. As the dataset is generated by the nature, the selected class subset is no longer generated by Grassmannian packing or the isotropic distribution as assumed in the theoretical July 31, 2020 DRAFT P = 0.01 P = 0.05 P = 0.25 ⋅ ⋅ ⋅  analysis. However, we can still observe the capacity gain of class selection from Fig. 4, e.g., 33% capacity gain at the communication rate of 10 5 bit/s. Furthermore, the capacity with class selection scales with a growing communication rate at a rate faster than the random-class case.
Both trends are aligned with the analytical results.

VII. CONCLUDING REMARKS
In this work, we have studied the performance of remote classification over wireless channels.
The main contribution is the establishment of a relation between classification and communication by proposing various metrics of classification capacities and analyzing them using tools from differential geometry. This has led us to discover that the freedom of choosing object classes for classification under the channel constraint can attain an exponential scaling law of classification capacity w.r.t. the communication rate; without a deliberate selection, the scaling is linear.
The current study opens numerous directions for further investigation. Several of them are particularly interesting, including a realistic latency model, use of advanced wireless techniques (e.g., MIMO and OFDM) to increase the classification capacity, as well as the design of multiuser remote classification system that gives rise to new issues in terms of, e.g., resource allocation and cooperation.

A. Proof of Lemma 2
First, we prove the upper bound on P (i → j). As ω 2 + a 2 k ≥ a 2 k , ∀ω, it follows from (12) that On the other hand, one can easily verify that The above results suggest that, given d 2 i,j = K k=1 sin 2 θ (i,j) k , P ub is maximized when as many principal angles as possible are equal to zero. Consequently, one can further bound (39) as Next, we prove the lower bound on P (i → j). By Lemma 1, Similarly, following the same argument as before with is minimized if all the principal angles have the same value given d 2 i,j = K k=1 sin 2 θ (i,j) k . This leads to: This completes the proof.

B. Proof of Theorem 1
First, we prove the lower bound on the -classification capacity. By (18), For a high data SNR, it follows from the above equation that L 2 N 2 K log 2 K+K log 2 log 2 (1+g(σ 2 s ))−K log 2 log 2 1+g(σ 2 s ) 2 . (47) Next, we prove the upper bound on the -classification capacity. From (19), As the direct approach is intractable, we find a lower bound on the right-hand side of (48).
Using the fact that 4KL − 2 N K ≥ δ 2 ub > 1 for large N and K, (50) The substituting of N = βR gives (20). Furthermore, as R, K → ∞, the bounds on theclassification capacity scale in (21), which completes the proof.
Combining the above result with (51) and (52), (22) follows. Following the same procedure, the upper bound can be proved.
This implies that R δ scales linearly with log 2 ρ given δ. As a result, as ρ → ∞, R δ → ∞. Then, by letting R δ , K → ∞, both bounds scale as shown in (26), which completes the proof.

E. Proof of Lemma 6
To derive the upper bound on the CDF, namely F d 2 c (x), we fist obtain an upper bound on the probability of Pr(sin 2 θ max < x). Given (27), Due to the fact that Pr(sin 2 θ max < x) = Pr(θ max < arcsin √ x), one can have where the inequality uses the fact that F 2 1 is a non-decreasing function in x. On the other hand, according to [29], one can have Then, by substituting (58) into (57), Pr(sin 2 θ max < x) = Pr(θ max < arcsin x) ≤ x As two random subspaces of dimension K embedded in R N are quasi-orthogonal, given large N , the squared chordal distance d 2 c can be approximated as K sin 2 θ max . Then, we can bound the said CDF, namely F d 2 c (x), as F d 2 c (x) = Pr(d 2 c < x) ≈ Pr(K sin 2 θ max < x) = Pr sin 2 θ max < x K , N → ∞.