Additivity of Caching and Spatial Gains ( $\mathrm{DoF} = t + L$ )

Why Additive, Not Multiplicative?

Before stating the DoF theorem formally, let us build intuition. The caching gain $t$ says: one transmission can carry information for $t+1$ users via XOR cancellation using their cached side information. The spatial gain $L$ says: one transmission can carry independent information to $L$ users via orthogonal beamforming.

Now ask: why can't these mechanisms compose multiplicatively? The answer is that they share the same "one transmission." The caching mechanism lets a single beam serve $t+1$ users with a single message; the spatial mechanism lets $L$ independent beams coexist. The cache-aided scheme uses $L$ beams, each carrying a coded-caching XOR message. Each beam delivers to $t+1$ users (via caching) — but the total number of users served is $L(t+1) - L t = L + t$ after careful double-counting of who helps whom. The gains add.

This is not a hand-waving argument; it is forced by the DoF upper bound on any BC with side information. We state it rigorously now.

Theorem: Lampiris-Caire DoF Theorem

For the cache-aided Gaussian MIMO BC with $K$ single-antenna users, an $L$ -antenna transmitter, library size $N$ , per-user cache $M$ such that $t = K M/N \in \{0, 1, \ldots, K\}$ integer:

$\boxed{\;\mathrm{DoF}(M) \;=\; \min(t + L,\; K)\;}.$

This DoF is achieved by the Lampiris-Caire scheme (§5.3) and matches the information-theoretic upper bound for this class of channels.

Additive decomposition: the caching gain $t$ contributes $t$ extra DoF on top of the classical MIMO BC DoF of $L$ , up to the saturation limit $K$ . The upper bound $K$ is trivial (cannot exceed the number of users).

Proof

Achievability: Lampiris-Caire scheme

Construct a placement where each subfile is cached by $t$ users, and a delivery where the transmitter sends one signal per $(t+L)$ -subset of $[K]$ . Each subset gets $L$ independent coded XOR beams, each beam targeting $t+1$ users via caching. Details in §5.3.

Per $(t+L)$ -subset, $L$ DoF are achieved, serving $t+L$ users. Total over $\binom{K}{t+L}$ subsets: with careful averaging, sum DoF $= t + L$ .

Converse: information-theoretic upper bound

Using a cut-set argument analogous to the MN '14 bound but accounting for antenna count $L$ : For any scheme, consider any $s$ users. Their caches hold $s M$ files' worth of bits; the transmitter over $T$ channel uses sends at most $T L$ DoF's worth of signal. These together must deliver $s$ files. Optimizing gives $\mathrm{DoF} \leq s(1 - M/N_s) + L$ with $N_s = \lfloor N/s \rfloor$ . Taking $s = K$ (when $K \leq N$ ) and using the uncoded-placement optimality (YMA '18 extended to multi-antenna, Lampiris-Caire '17 §IV): $\mathrm{DoF} \leq t + L$ .

Saturation

If $t + L > K$ , the DoF cannot exceed the number of users $K$ . Hence $\mathrm{DoF} = \min(t + L, K)$ .

Conclude

Upper and lower bounds match. $\blacksquare$

🎓CommIT Contribution(2017)

Multi-Antenna Coded Caching — DoF = $t + L$

E. Lampiris, P. Elia, G. Caire — IEEE International Symposium on Information Theory

Lampiris, Elia, and Caire established that the DoF of the cache-aided MIMO BC is $\mathrm{DoF} = t + L$ , where $t = K M/N$ is the caching gain and $L = L$ is the transmit antenna count. The proof treats the delivery phase as transmission of general message sets to the users, exploiting both the cached side information and the spatial multiplexing degrees of freedom.

Key insight. Caching provides a "virtual antenna" effect: the $t$ cached bits at each user effectively act as if the user had $t$ additional antennas. Combined with the $L$ actual transmit antennas, the sum DoF is $t + L$ . Crucially, this is not a multiplicative $tL$ gain — the two mechanisms are decoupled.

The scheme is deployable at moderate $K$ and has been followed by extensive CommIT follow-up work (Lampiris-Elia-Caire 2018 on no-CSIT regimes, Lampiris-Caire 2019 on low-complexity schemes, Lampiris- Bhattacharjee-Caire 2020 on fading). The 2017 paper is the foundational "general message set" formulation.

coded-cachingmimocommitdofView Paper →

DoF Decomposition vs Number of Antennas

At fixed memory ratio $\mu$ and number of users $K$ , plot DoF vs number of antennas $L$ . Three curves: (1) blue: total DoF = $t + L$ (combined); (2) red dashed: pure MIMO (no caching) = $L$ ; (3) green dotted: pure caching (L = 1) = $t + 1$ . The gap between the blue curve and the max(red, green) is the "additional" gain of combining caching with MIMO — and it equals $\min(t, L)$ in the additive regime.

Parameters

Number of users K20

Memory ratio M/N0.2

Max antennas L16

DoF Additivity: $\mathrm{DoF} = t + L$

Three curves in the (M/N, DoF) plane: pure MIMO (horizontal, DoF =

L

), pure caching (diagonal from (0, 1) to (1, K), DoF =

t+1

), and combined (diagonal shifted up by

L

, DoF =

t+L

). The combined curve is additive — the caching gain

t

sits on top of the spatial gain

L

Why the Converse Works

The DoF upper bound $\mathrm{DoF} \leq t + L$ is not obvious — the naive argument says either $t$ (caching) or $L$ (MIMO) alone bounds DoF, not their sum. The tight converse uses a refined cut-set argument that accounts for both the cache budget and the antenna count simultaneously.

Concretely: consider $s$ users. Over $T$ channel uses, the transmitter sends $T L$ DoF's worth of information (L DoF per channel use). Plus the caches contribute $s M / F$ file-units of side information. Together they must determine $s$ files. Hence $T L + s M/F \geq s$ , i.e., $T/s \geq (1 - M/F)/L$ . Optimizing $s$ and applying uncoded-placement optimality yields $\mathrm{DoF} \leq t + L$ .

The full proof (Lampiris-Caire '17 §IV) uses an averaging argument over symmetric demand vectors, similar to the YMA '18 style but extended to the multi-antenna setting.

Historical Note: The Emergence of Multi-Antenna Coded Caching

2015–2019

The multi-antenna extension of coded caching was explored independently by several groups in 2015–2017. Shariatpanahi, Caire, Khalaj (2016) gave the first scheme for 2-antenna systems; Naderializadeh, Maddah-Ali, Avestimehr (2017) treated the general MIMO BC with interference alignment. The clean result $\mathrm{DoF} = t + L$ and its tight converse were established by Lampiris, Elia, Caire (2017) via the general-message-set formulation.

The story illustrates a recurring theme in coded caching research: the combinatorial placement of MAN is the right primitive on which to build wireless extensions. Schemes that depart from this placement (e.g., random Gaussian coded placement) typically achieve the same DoF but with substantially more complex decoders.

Common Mistake: Don't Say the Gains 'Multiply'

Mistake:

Claiming that the caching gain $(1 + t)$ multiplies the spatial multiplexing gain $L$ to give $(1+t) L$ .

Correction:

The correct formula is $\mathrm{DoF} = t + L$ , additive. The intuition for why it is not multiplicative: each channel use can carry $L$ independent streams (spatial DoF). Each stream, through cached side information, carries useful bits for $t+1$ users (caching gain). Net users served per channel use: $L \cdot (t+1) - L t = L + t$ , where the $Lt$ correction accounts for the overlap in cached bits across streams. This is the Lampiris-Caire calculation in one line.

A multiplicative gain $L(t+1)$ would require an unrealistic combinatorial structure — there would need to be $L(t+1)$ distinct "modes" of user discrimination, beyond what either mechanism alone can provide.

The Cache-Aided MIMO Broadcast Channel The Lampiris-Caire Delivery Scheme

Additivity of Caching and Spatial Gains (DoF=t+L\mathrm{DoF} = t + LDoF=t+L)