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Cache-Aided Interference Channels

Mohammad Ali Maddah-Ali, Urs Niesen

arXiv:1510.06121v2cs.IT

TL;DR

The paper asks how caching can improve the wireless link in interference channels, beyond reducing backhaul load. It formulates placement and delivery for cached transmitters, focusing on three users, and proposes a scheme combining load balancing, interference cancellation, and interference alignment. For three users, the scheme achieves the stated cache-size/reciprocal-DoF bound and jointly uses these three gains.

  • Problem

    The paper studies how caching can benefit wireless communication between caches and end users, rather than only reducing backhaul and core-network load.

  • Method

    The paper jointly designs cache placement and content delivery in a three-user interference channel with isolated transmitter caches.

  • Results

    For three users, the proposed scheme achieves the stated piecewise upper bound on 1/DoF(µ) while exploiting load balancing, interference cancellation, and increased interference alignment.

  • Takeaways & Limitations

    Caching can provide distinct wireless gains by balancing transmitter load, enabling interference cancellation, and increasing interference-alignment opportunities.

Abstract

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Over the past decade, the bulk of wireless traffic has shifted from speech to content. This shift creates the opportunity to cache part of the content in memories closer to the end users, for example in base stations. Most of the prior literature focuses on the reduction of load in the backhaul and core networks due to caching, i.e., on the benefits caching offers for the wireline communication link between the origin server and the caches. In this paper, we are instead interested in the benefits caching can offer for the wireless communication link between the caches and the end users. To quantify the gains of caching for this wireless link, we consider an interference channel in which each transmitter is equipped with an isolated cache memory. Communication takes place in two phases, a content placement phase followed by a content delivery phase. The objective is to design both the placement and the delivery phases to maximize the rate in the delivery phase in response to any possible user demands. Focusing on the three-user case, we show that through careful joint design of these phases, we can reap three distinct benefits from caching: a load balancing gain, an interference cancellation gain, and an interference alignment gain. In our proposed scheme, load balancing is achieved through a specific file splitting and placement, producing a particular pattern of content overlap at the caches. This overlap allows to implement interference cancellation. Further, it allows us to create several virtual transmitters, each transmitting a part of the requested content, which increases interference-alignment possibilities.

I. INTRODUCTION

The paper reformulates interference-channel communication for centrally generated content that can be cached before transmission. In the three-user setting, joint placement and delivery exploit load balancing, interference cancellation, and interference alignment.

  • Motivation: Content-centric wireless traffic makes the classical assumption of independent transmitter messages questionable.Content is centrally generated and can be available at multiple network locations before transmission.
  • Problem shift: The proposed formulation separates content placement from delivery and requires transmitters to satisfy arbitrary receiver demands using cached message functions.Placement occurs before demands are known; delivery uses the cached content to serve requested messages.
  • Contributions: The three-user framework provides a placement and delivery scheme designed to realize three distinct caching gains.The gains are load balancing, interference cancellation, and interference alignment.
  • Caching gains: Load balancing distributes delivery load across transmitters, avoiding bottlenecks.
  • Caching gains: Content overlap across transmitters enables transmit zero forcing to cancel requested-content interference at unintended receivers.
  • Caching gains: Proper placement increases interference-alignment opportunities, while combining alignment with zero forcing requires handling dependent effective channel coefficients.The scheme addresses this dependence using additional precoding factors and the algebraic mapping to original channel coefficients.

II. PROBLEM FORMULATION

The paper models cached content delivery over a K-user Gaussian interference channel, with cache placement preceding demand-aware delivery. It characterizes the cache-size/sum-DoF tradeoff under explicit storage, channel, power, and side-information assumptions.

  • Channel assumptions: The channel is a K-user Gaussian interference channel with known time-invariant gains, complex Gaussian receiver noise, and an average power constraint P.
  • System model: The system has N independent files, and each transmitter stores MF bits, equivalently M entire files, in a local cache.The normalized cache size is the fraction of the file library that each transmitter can store locally.
  • Two-phase operation: Communication has a placement phase using the full library and a delivery phase in which receivers request files.Transmitters then encode using only their local caches together with the revealed demands and channel coefficients.
  • Performance criterion: Achievable rate requires vanishing error as file size grows, and capacity is the supremum of rates satisfying this condition.The error criterion maximizes over all demands and receivers.
  • Objective: The objective is to characterize the tradeoff between normalized cache size µ and sum degrees of freedom DoF(µ).The reciprocal 1/DoF(µ) is used because it is convex in µ under memory sharing.
  • Information timing: Cache contents cannot depend on future demands or channel gains because this side information is revealed only during delivery.Placement and delivery occur at different times and network-load conditions.
  • Domain: The meaningful cache-size domain is 1/K ≤ µ ≤ 1: below 1/K not all library bits can be cached, while µ ≥ 1 gives no further benefit.

III. MAIN RESULTS

For three users, the paper gives an achievable cache-aided interference-channel scheme and expresses the resulting bound through reciprocal sum degrees of freedom. The construction combines load balancing, interference cancellation, and increased interference alignment.

  • Main result: K = 3 is the focus, and the paper presents an achievable scheme yielding a lower bound on DoF(µ).Results are expressed using 1/DoF(µ), whose convexity supports the analysis.
  • Main result: 13/18 − µ/2 for 1/3 ≤ µ ≤ 2/3, and 1/2 − µ/6 for 2/3 ≤ µ ≤ 1, upper-bound 1/DoF(µ).The stated bound holds for almost all channel gains H ∈ C^3×3.
  • Main result: The upper bound on reciprocal degrees of freedom is depicted as a tradeoff between normalized cache size and 1/DoF(µ).
  • Achievability: Convexity allows achievable corner points at µ = 1/3, 2/3, and 1 to generate achievable points on connecting lines.Detailed proofs are provided separately after the outline.
  • Caching gains: The proposed scheme exploits load balancing, interference cancellation, and increased interference alignment simultaneously.

A. Corner Point at µ = 1

With µ = 1, every transmitter caches the entire content library, enabling cooperative delivery of distinct requested files. This cooperation achieves interference cancellation through transmitter zero-forcing without additional delivery-phase backhaul collaboration.

  • A. Corner Point at µ = 1: Each transmitter caches the entire content library when µ = 1, so all requested files are locally available at every transmitter.Receivers request files A, B, and C, and each transmitter can access all three files.
  • A. Corner Point at µ = 1: Cooperative transmission uses multiple-antenna broadcast techniques, including zero-forcing, to achieve a sum DoF of 3.
  • A. Corner Point at µ = 1: Cache-enabled cooperation requires no backhaul collaboration during delivery and performs interference cancellation through transmitter zero-forcing.
  • A. Corner Point at µ = 1: Realizing the zero-forcing gain in practice still involves system-level challenges beyond the theoretical derivation.

B. Corner Point at µ = 1/3

At µ = 1/3, each transmitter stores a distinct third of every file, allowing requested files to be delivered partially from all transmitters. Compared with whole-file placement, this balances transmitter load and increases interference-alignment opportunities, achieving sum DoF 9/5 in the discussed scheme.

  • B. Corner Point at µ = 1/3: At µ = 1/3, transmitters collectively store the entire library, with each transmitter caching one third of the content.
  • B. Corner Point at µ = 1/3: Each file is split into three equal nonoverlapping subfiles, and transmitter k caches subfile W_n,k for every file.
  • B. Corner Point at µ = 1/3: With requests A, B, and C, each requested file is distributed across the three transmitters as corresponding subfiles.
  • B. Corner Point at µ = 1/3: Whole-file placement can bottleneck the system: when all requested files reside at one transmitter, occurring for 1/9 of requests, the sum DoF is limited to 1.
  • B. Corner Point at µ = 1/3: When each requested file resides at a distinct transmitter, whole-file placement forms a standard interference channel with sum DoF 3/2.
  • B. Corner Point at µ = 1/3: The proposed placement avoids bottlenecks for all receiver requests and increases alignment opportunities because each requested file can be partially delivered from every transmitter.The proposed scheme achieves sum DoF 9/5, exceeding the cited alternative cases.

C. Corner Point at µ = 2/3

At normalized cache size µ = 2/3, files are split and placed so transmitter overlap supports cooperative zero forcing and creates virtual transmitters for interference alignment. The resulting scheme achieves per-user DoF 6/7 and sum DoF 18/7.

  • Content placement: Each file is split into three equal subfiles labeled by transmitter pairs, so every subfile is cached at two of the three transmitters.For example, A = (A12, A13, A23), with each subscript identifying the caching transmitters.
  • Interference cancellation: Overlapping cache contents let the two transmitters holding each subfile cooperate and cancel its interference at a selected unintended receiver.They transmit along a direction orthogonal to that receiver’s channel vector, producing zero at the targeted receiver.
  • Virtual transmitters: Zero forcing transforms the original channel into an equivalent channel with 18 virtual transmitters, each carrying one file part.The equivalent channel preserves nonzero gains toward intended receivers while eliminating designated interference links.
  • Delivery design: For demand (A, B, C), each cached subfile is further split into two equal parts, each designated for zero forcing at one unintended receiver.For example, A12 = (A2 12, A3 12), where the superscript identifies the receiver to be zero-forced.
  • Interference alignment: Each receiver aligns six remaining interference terms while decoding six desired file parts, achieving per-user DoF 6/7 and sum DoF 18/7.With each file part carrying 1/7 DoF, aligned interference uses 1/7 DoF and desired content uses 6/7 DoF at each receiver.
  • Alignment challenge: Real interference alignment is nontrivial because 36 equivalent-channel coefficients depend on only nine original coefficients, but scaling and adjugate-based analysis restore feasibility.The scheme applies a distinct scaling prefactor to each equivalent input and uses the algebraic structure of the adjugate operator.

A. Preliminaries

The preliminaries review real and complex interference alignment, using analytic monomial mappings and Diophantine approximation to construct decodable signal constellations. The resulting minimum-distance and degrees-of-freedom properties support reliable high-SNR communication.

  • Real interference alignment is reviewed through analytic maps and their complex-channel extension.The framework uses analytic functions whose values determine the signal construction.
  • Theorem 3 gives ω(f(h)) = I/2 −1 for almost all h when 1, f1, …, fI−1 are linearly independent analytic functions.This number-theoretic result supplies the Diophantine property needed for minimum-distance analysis.
  • The constellation construction uses integer points in [−Q, Q] and analytic mappings scaled for a per-symbol power constraint P.The maps are instantiated as monomials of channel coefficients.
  • The minimum distance satisfies ∆ ≥ c2 and grows as c1c2P^ε/2 under the stated analytic-function assumptions.The constants c1 and c2 control scaling and account for the finite exceptional set, while c2 is positive for almost all h.
  • Each integer-valued term contributes log(2Q + 1) bits and a DoF approaching 1/I as ε becomes small.The growing minimum distance makes the average probability of error arbitrarily small at high signal-to-noise ratios.

B. Achievable Scheme

The achievable scheme splits requested files into cache-overlapping parts, creates virtual transmitters, zero-forces selected interference, and aligns the remaining streams. For the three-user channel, the construction achieves a per-user DoF approaching 6/7 and sum DoF arbitrarily close to 18/7.

  • Content placement and virtual transmitters: Each file is split into six subfiles, and each file part is stored at two transmitters and zero-forced at one unintended receiver.This placement creates virtual transmitters for individual file parts.
  • Interference cancellation: Zero forcing uses scaling factors hτ˜k and −hτk, making the equivalent channel coefficient zero at the targeted receiver.At other receivers, the resulting coefficient is a structured combination of the original channel gains.
  • Equivalent channel: The construction produces 18 virtual transmitters; each receiver decodes six desired virtual-transmitter streams while six others are zero-forced and six remain interfering.The remaining interference is handled through alignment.
  • Interference alignment: Each group of six interfering streams is formed from L6 integer-modulated substreams scaled by distinct monomials of TL(u^(1)).The monomial construction is applied symmetrically across receivers and interfering file parts.
  • Interference alignment: The received interference constellations collapse into a smaller constellation than the product of the six transmitted cardinalities, establishing interference alignment.The construction relies on the monomial definition and the chosen channel-dependent scaling factors.
  • Decodability: Linear independence and Theorem 3 guarantee decodeability for almost all channel matrices with fixed scaling factors.The adjugate mapping preserves measure-zero exceptional sets, extending the result from adjugate matrices to channel matrices.
  • Achievable degrees of freedom: The scheme achieves per-user DoF approaching 6/7 and sum DoF arbitrarily close to 18/7 as L grows and ε tends to zero.The minimum distance grows with P, so the probability of error tends to zero as power increases.

V. CONCLUSIONS AND DISCUSSION

The paper formulates cache-aided interference communication for three transmitters and receivers by combining transmitter zero forcing with interference alignment. Its discussion shows that the relative value of the two gains depends on cache size, while jointly achieving both gains beyond three transmitters remains open.

  • The paper proposes and analyzes a three-transmitter, three-receiver scheme combining transmitter zero forcing with interference alignment.The analysis addresses dependence among effective channel coefficients introduced by zero forcing.
  • Follow-up work extends the setting to arbitrary receivers with three transmitters and adds delivery-phase backhaul constraints.These developments broaden the model beyond the original setting.
  • Using only transmitter zero forcing or only interference alignment provides alternative benchmarks for the same specialized setting.The paper compares its joint scheme against both single-technique approaches.
  • For small cache sizes alignment supplies the main gain, for large cache sizes zero forcing supplies the main gain, and moderate sizes require both.Figure 6 compares inverse degrees of freedom across the joint and single-technique schemes.
  • Jointly achieving zero-forcing and alignment gains with more than three transmitters remains an open problem.

APPENDIX A PROOF OF LEMMA 1

The appendix proves convexity of reciprocal degrees of freedom by combining coding schemes through file splitting and memory partitioning. This establishes convexity of 1/DoF(µ) as a function of normalized cache size.

  • The combined scheme has vanishing error probability as file size F tends to infinity when both split proportions are positive.This follows because each component scheme has vanishing error on its own reduced file size.
  • Two coding schemes with normalized cache sizes µ1 and µ2 can be combined by splitting each file and each cache into corresponding parts.Each component scheme operates on its assigned file subfiles with its original normalized cache size.
  • The combined rate is obtained by dividing the total file size by the sum of the channel uses required by the two component schemes.Scheme i uses αiF/Ri channel slots for its assigned αiF file bits.
  • The achievable memory–reciprocal-rate region A(P) is convex for fixed power constraint P.The construction yields convex combinations of memory–reciprocal-rate pairs.
  • Therefore, reciprocal degrees of freedom 1/DoF(µ) is convex in normalized cache size µ.

APPENDIX B THE ADJUGATE AND SETS OF MEASURE ZERO

The appendix defines the adjugate and establishes that measure-zero sets remain measure zero under its preimage, including for non-invertible matrices. The proof uses the adjugate’s bijectivity and continuous differentiability on invertible matrices.

  • Adjugate definition: The adjugate entry (i, j) of A is (−1)i+j det(Mj,i), where Mj,i removes row j and column i.This definition applies to matrices A ∈ C^K×K.
  • Extension to all matrices: For any B ⊂ C^K×K, including sets containing non-invertible matrices, λ(B) = 0 implies λ(adj−1(B)) = 0.The appendix extends the invertible-case argument to the full matrix space by considering the measure-zero non-invertible set separately.
  • Invertible-matrix restriction: For invertible matrices, adj(·) is bijective on GL, and its inverse is continuously differentiable.The appendix first restricts the adjugate to GL, the open set of invertible matrices.
  • Measure-zero preservation: If B ⊂ GL has Lebesgue measure zero, then adj−1(B) also has Lebesgue measure zero.This follows from the continuous differentiability of adj−1 on the open set GL.
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