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Interference Alignment as a Rank Constrained Rank Minimization
Dimitris S. Papailiopoulos, Alexandros G. Dimakis
TL;DR
The paper addresses the open problem of maximizing spatial degrees of freedom in static flat-fading MIMO interference channels beyond a few special cases. It reformulates interference alignment as rank-constrained rank minimization, relaxes rank using normalized nuclear norms, and reports strong performance across many setups.
Problem
Maximizing the available spatial degrees of freedom remains open beyond a few special cases, motivating a formulation of interference alignment for limited dimensions.
Method
The paper characterizes interference alignment through signal and interference spaces, reformulates it as rank-constrained rank minimization, and uses a normalized sum of nuclear norms as a convex relaxation.
Results
The proposed scheme is sum-DoF optimal for many setups where perfect interference alignment is possible, and can provide extra degrees of freedom over leakage minimization in some cases.
Takeaways & Limitations
The rank-based framework provides a tractable approach to interference alignment that can outperform leakage minimization when perfect alignment is not attained.
Abstract
from arXiv · showhide
We show that the maximization of the sum degrees-of-freedom for the static flat-fading multiple-input multiple-output (MIMO) interference channel is equivalent to a rank constrained rank minimization problem (RCRM), when the signal spaces span all available dimensions. The rank minimization corresponds to maximizing interference alignment (IA) so that interference spans the lowest dimensional subspace possible. The rank constraints account for the useful signal spaces spanning all available spatial dimensions. That way, we reformulate all IA requirements to requirements involving ranks. Then, we present a convex relaxation of the RCRM problem inspired by recent results in compressed sensing and low-rank matrix completion theory that rely on approximating rank with the nuclear norm. We show that the convex envelope of the sum of ranks of the interference matrices is the normalized sum of their corresponding nuclear norms and introduce tractable constraints that are asymptotically equivalent to the rank constraints for the initial problem. We also show that our heuristic relaxation can be tuned for the multi-cell interference channel. Furthermore, we experimentally show that in many cases the proposed algorithm attains perfect interference alignment and in some cases outperforms previous approaches for finding precoding and zero-forcing matrices for interference alignment.
I. INTRODUCTION
The paper reformulates interference alignment in static flat-fading MIMO channels as rank-constrained rank minimization and develops a nuclear-norm relaxation to maximize spatial DoF. Experiments indicate sum-DoF optimality in many feasible cases, with gains and robustness over leakage minimization in some settings.
- Motivation: Interference alignment seeks to maximize interference-free signaling dimensions by designing transmit precoding and receive zero-forcing matrices.For static flat-fading MIMO channels with perfect channel knowledge, the design flexibility is concentrated in these matrices.
- Motivation: NP-hard matrix construction, few closed-form solutions, and difficult feasibility characterization make perfect interference alignment challenging.The difficulty is attributed to the problem’s over-constrained nature.
- Rank formulation: The paper characterizes alignment by imposing full-rank useful-signal constraints while minimizing the ranks of interference spaces.Full-rank constraints preserve all available spatial dimensions, whereas rank minimization collapses interference into the smallest possible subspaces.
- Rank formulation: Under those constraints, minimizing the sum of interference-matrix ranks is equivalent to maximizing the sum of spatial DoF.This formulation motivates a natural approximation despite being harder to solve exactly than bilinear perfect-IA equations.
- Convex relaxation: The proposed convex heuristic minimizes a normalized sum of interference nuclear norms and approximates full-rank constraints with positive minimum-eigenvalue conditions.The nuclear norm corresponds to the ℓ1-norm of singular values, favoring sparse singular-value solutions rather than merely low-energy ones.
- Evaluation and extensions: Experiments find sum-DoF optimality in many setups where perfect alignment is possible, plus extra DoF and robustness to singularities in some cases.The comparison is against leakage minimization, including diagonal channel structures where that approach can encounter singularities.
- Evaluation and extensions: The relaxation extends to K-cell interference channels by adding tractable affine constraints on the precoding matrices.The same rank-constrained rank-minimization perspective motivates the convex programming formulation for structurally constrained beamforming.
II. SYSTEM MODEL
The system is a static flat-fading K-user MIMO interference channel in which each user transmits d symbols through linear precoding and each receiver applies a zero-forcing filter. Signal spaces are modeled alongside interference spaces, with assumptions that simplify antenna and stream dimensions while allowing unequal antenna counts.
- Each of K transmitters has Mt transmit antennas, each receiver has Mr receive antennas, and users synchronize transmissions.
- User k transmits a d-dimensional symbol vector after linear precoding, where d is the pursued degrees of freedom per user.Achievable DoF correspond to signal-space dimensions free of interference.
- Receiver k applies a linear zero-forcing filter Uk with d linearly independent columns to obtain the useful signal space.
- The model distinguishes useful signal spaces from the space spanned by horizontally concatenated interference matrices.
- The channel is denoted an (Mr × Mt, d)K system, assuming signal spaces span all available dimensions and commonly equal antenna counts and stream demands.The formulation can be extended to transmitters and receivers with different numbers of antennas.
IV. A NUCLEAR NORM HEURISTIC
The paper formulates sum-DoF maximization as rank minimization of interference spaces subject to full-rank useful signal spaces, then relaxes both the objective and constraints convexly. An alternating nuclear-norm procedure targets low-rank interference while preserving useful dimensions, but converges only to a local optimum and lacks straightforward exact-solution guarantees.
- Maximizing sum spatial DoF is equivalent to an RCRM problem that minimizes interference-matrix ranks while enforcing useful-signal rank constraints.The formulation applies to static flat-fading MIMO interference channels.
- The nuclear norm, equal to the ℓ1-norm of singular values, provides the convex envelope of the normalized sum of interference ranks.Low-rank interference is the desired outcome because it confines interference to fewer dimensions.
- Alternating leakage minimization may produce low-energy interference without minimizing its span, and can yield rank-deficient useful signal spaces with zero or very low DoF.The proposed positivity constraint explicitly preserves full-rank signal spaces across channel structures.
- The algorithm alternates between optimizing precoding and zero-forcing matrices under the convex relaxation after arbitrarily selecting one matrix set.The procedure iterates these updates and is bound to converge to a local optimum.
- The nuclear-norm approximation experimentally favors low-rank interference solutions and often attains perfect IA, while exact rank-minimization guarantees do not transfer straightforwardly.
- Full-rank useful-signal constraints are approximated with positive-semidefinite signal-space matrices whose minimum eigenvalues are constrained positively.As the eigenvalue threshold ϵ decreases, the relaxed feasible sets asymptotically overlap the corresponding open full-rank condition.
V. INTERFERENCE ALIGNMENT FOR CELLULAR NETWORKS
For K-cell interference channels, the paper adapts the MIMO formulation by imposing affine zero-entry constraints on precoding matrices that encode which users transmit which symbols. The resulting approximation retains the nuclear-norm framework and produces low-rank interference solutions associated with high sum-rate.
- A K-cell channel is represented as a general K-user MIMO interference channel in which each symbol is transmitted from a specified subset of antennas.
- Each cell supports d users, with user u transmitting one symbol through a beamforming vector vk,u.
- The cellular approximation adds affine constraints that force appropriate precoding-matrix entries to zero.These constraints encode the antenna or transmitter structure of the cellular channel.
- No further approximation is required for the cellular structure because affine constraints are tractable.
- The cellular approximation yields low-rank interference matrices, resulting in high sum-rate in the reported simulations.
A. Interference Channel
The evaluation simulates several proper MIMO and symbol-extended interference-channel systems, averaging sum-rate and interference-free dimensions over random channel realizations. Interference-free dimensions are estimated from singular-value thresholds, while the reported metric does not capture low-SNR rate slope.
- The simulations include (4×8, d = 1, 3)3, (6×6, d = 1, 3)3, a 3-user symbol-extended single-antenna channel, and a (4 × 18, d = 1, 2)10 system.
- All considered MIMO systems are proper, satisfying d ≤ Mt+Mr over K+1.
- The evaluation plots each system’s sum-rate and average number of interference-free dimensions per user.
- Interference-free dimensions are counted using singular values of Sk and Jk above 10^-6, but this metric omits the low-SNR rate slope.
1) 3-user interference channel:
Across the 3-user simulations, performance depends on SNR and stream dimension: max-SINR often leads at lower SNR, while the proposed method gains at higher SNR or larger d.
- Perfect interference alignment is expected to be feasible with high probability because the simulated systems are proper.
- For d = 1, max-SINR outperforms the proposed and leakage-minimization methods at low to moderate SNRs.
- The nuclear-norm method obtains more than 2 DoF for d = 3, whereas max-SINR and leakage minimization do not exceed 1 per-user DoF.
3) 10-user interference channel:
In the 10-user channel, the proposed method is competitive at high SNR for d = 1 and can gain rate from extra DoF, but leakage minimization performs better for d = 2 in the tested range.
- For d = 1, max-SINR leads at low to moderate SNR, while the proposed and max-SINR rates converge and slightly exceed leakage minimization later.
- Past 40dB, leakage minimization and max-SINR with QR exhibit zero-slope sum-rate curves, while extra DoF from the proposed method provide slightly better high-SNR performance.
B. Cellular Interference Channel
In the cellular channel, the proposed method improves substantially over random beamforming and interference zero-forcing as iterations increase, reaching roughly three times the sum-rate at 60dB in one comparison.
- The experiment uses a 3-cell channel with 2 users per cell, 3 transmit antennas, and 4 receive antennas.
- The proposed method is compared with random beamforming and an interference-zero-forcing scheme over 200 channel realizations and powers from 0 to 60dB.
- One iteration is comparable to the baselines, while two iterations produce approximately twice the sum-rate at 60dB and ten iterations approximately three times more.
- Across all tested power configurations, the proposed algorithm outperforms random beamforming and interference zero-forcing.
VII. CONCLUSION
The paper reformulates interference alignment as rank constrained rank minimization and relaxes both objective and constraints using nuclear-norm-based convex techniques. It reports empirical success but leaves tightness guarantees as an open problem.
- The interference-alignment problem is reformulated as rank constrained rank minimization.
- The framework introduces individually tight convex relaxations for the RCRM objective and constraints, inspired by nuclear-norm rank relaxation.
- The paper does not establish theoretical guarantees for when the nuclear-norm relaxation is tight.