Discretization-Based Lifting in Networks

• Discretization-based lifting is a method that approximates complex Gaussian networks with deterministic surrogate models to simplify code design and capacity analysis.
• It employs quantization and structured codebook pruning based on typicality to ensure that codes remain reliable under Gaussian noise.
• The approach guarantees a bounded, SNR-independent rate penalty determined by network size and antenna configuration, making it applicable to relay, multicast, and MIMO networks.

Discretization-based lifting is a family of methodologies in which a continuous, high-dimensional, or otherwise intractable mathematical object or process is approximated using discretization, and then “lifted” to a desired setting—typically for purposes of code or model design, algorithm initialization, or structural transformation. In telecommunications and information theory, discretization-based lifting is especially central to the development of digital interfaces and coding schemes, as exemplified by the discrete superposition model and its lifting to Gaussian relay networks. At its core, the approach enables near-optimal design in complex systems, leveraging deterministic (noise-free) surrogate models that admit explicit or algorithmically tractable constructions, and then employing lifting and pruning strategies to transfer these constructions into the original, noise-perturbed or more general domain while controlling losses in performance.

1. Discrete Superposition Model as a Surrogate for Gaussian Networks

The discrete superposition model serves as a deterministic, quantized approximation of a Gaussian relay network. It quantizes the input signals, the channel gains (by retaining only their integer part), and the outputs, so that the received signal at each node is a sum of integer-weighted inputs, with the products and sums also quantized by discarding fractional parts. This stripping away of noise and fine-scale variability yields a model retaining the core superposition property of wireless channels while rendering capacity analysis and code design far more tractable. Importantly, in high-SNR regimes, the quantization effects are minor, and the discrete superposition model can approximate the capacity of the original Gaussian network to within a bounded number of bits, uniformly across all SNR levels.

The deterministic nature of the superposition network allows construction of codes with specified mappings and decoders, leveraging explicit finite alphabets. The analysis of cut-based mutual information in this surrogate model lays the foundation for code design that remains valid even when lifted back into the Gaussian (noisy) setting.

2. Lifting Codes via Pruning and Typicality

Lifting in this context denotes the process by which a codebook, together with its corresponding relaying/decoding protocol designed for the discrete superposition model, is transformed for use in the original Gaussian relay network. Since the deterministic network does not account for noise, the lifting process employs pruning of the codebook: only a subset of codewords is retained, selected based on the e-strong typicality of the received vectors at each node. This subset is chosen so that, with high probability, all nodes will observe strongly typical sequences under the Gaussian channel laws, despite random Gaussian noise.

At each relay or node, a selection step prunes the codewords that would yield received signals falling outside a pre-specified strongly typical set for the deterministic model. This procedure is iteratively applied at every node in the relay network, sequentially shrinking the codebook until a subset remains that is robust under the effects of noise in the Gaussian setting.

Formally, if $R$ is the code rate in the discrete superposition model, the rate achieved after lifting remains $R - M \cdot K$, where $M$ is the number of nodes and $K$ is a constant determined by network size but independent of the SNR and channel realizations. For MIMO configurations, $K$ additionally depends on the number of transmit/receive antennas.

3. Technical Formulations and Performance Gap

The relationship between the rates of the discrete superposition network and the lifted code for the Gaussian network can be bounded by a uniform, SNR-independent gap: $R_{\text{Gauss}} = R_{\text{Discrete}} - M \cdot K$ with explicit expressions for $K$ such as $K = \log(12M - 2) + 11$.

This rate reduction can be attributed to two main losses:

• The uncertainty introduced by noise (since in the deterministic model all outputs are noise-free).
• Quantization mismatches and the constraints imposed by the typicality-driven pruning, which ensure decoding reliability in the stochastic regime.

These losses are not cumulative in SNR or channel gain, but are determined only by (a) the number of network nodes and (b) the number of antennas (for MIMO cases).

4. Applications to Relay, Multicast, and MIMO Networks

Discretization-based lifting is not limited to two-terminal relay channels. The general methodology extends to:

• Relay networks with multiple relays and a single source-destination pair.
• K × K Gaussian interference networks.
• Multicast and MIMO relay/multicast/interference networks.

In each case, the deterministic (superposition) model provides a coding playground where near-capacity-achieving codes can be developed independently of noise statistics. The lifting procedure then certifies that these code designs remain near-optimal—up to a bounded, explicit constant penalty—when transferred to the corresponding full Gaussian networks.

As a consequence, the discrete superposition network becomes a de facto digital interface for operating real (noisy) Gaussian relay networks, as any code construction for the deterministic setting directly determines the set of allowed transmit symbols, relaying functions, and decoding rules for the physical channel.

5. Comparison with Traditional Gaussian Network Coding

Traditional code design for Gaussian relay networks faces challenges including high-dimensionality, noise-induced uncertainty, and the need for complex, SNR-dependent optimizations. In contrast, the discretization-based lifting approach decouples code design from noise, harnessing the structural tractability of deterministic models.

Key comparative insights:

• The rate loss is tightly bounded and SNR-independent, unlike certain Gaussian-network-specific strategies.
• Pruning may slightly reduce the codebook size but guarantees robustness by only admitting codewords that would remain decodable under any typical noise realization.
• The method enables the transfer of structured or combinatorial codes from a simplified setting, anticipating extensions to explicit structured code constructions with favorable algebraic properties (as emphasized for future work).

6. Implications, Limitations, and Future Research Directions

The methodology highlights both immediate and long-term research trajectories:

• There is an open direction in developing structured (possibly algebraic) codes within the discrete superposition model whose lifts yield further improved performance or reduced complexity in Gaussian networks.
• Alternative pruning strategies or new lifting paradigms could potentially shrink the constant rate gap further, narrowing the discrepancy between deterministic and stochastic models.
• The robustness and universality of pruning-based lifting for networks beyond single source-destination pairs—such as those with multiple sources, irregular timing, and interference-dominated topologies—remain an open area, especially regarding necessary modifications to facilitate interleaving, causality, or buffering.
• Critically, quantifying and reducing the explicit dependency of the constant gap $K$ on network parameters could bring deterministic approximations even closer to the theoretical limits of the underlying Gaussian networks.

7. Summary Table: Rate Relationship and Applicability

Model/Configuration	Rate Gap Formula	Applicability Domain
Relay network	$R_\text{Gauss} = R - M K$	Single source-destination, Relay, MIMO
Interference network ($K \times K$)	$K_I = 6K + \log(144K+1)$	Interference channels, MIMO, Multicast
MIMO extension	Depends on nodes/antennas	Multiple antennas per node

This table summarizes the main closed-form relationships and the broad applicability of discretization-based lifting beyond strictly relay networks.

---

Discretization-based lifting bridges deterministic and stochastic modeling in network information theory. By approximating the full stochastic (Gaussian) network with a quantized, noise-free deterministic surrogate, and transferring code constructions back to the Gaussian setting through codebook pruning and typicality arguments, it enables near-capacity-achieving operation with a rate penalty given by an explicit, SNR-independent constant. This approach facilitates simplified code design, generalizes to a wide variety of wired and wireless networks, and sets the stage for future work on structured codes, gap reduction, and extension to broader classes of communication networks (Anand et al., 2010, Anand et al., 2010).