InvCSINet: Invertible CSI Feedback for MIMO

• InvCSINet is an invertible CSI feedback framework for FDD massive MIMO that uses a bijective neural network to preserve information during compression.
• The architecture integrates adaptive quantization, differentiable bit-channel distortion, and latent information compensation to jointly optimize compression, transmission, and reconstruction.
• Evaluated on the DeepMIMO dataset, InvCSINet outperforms traditional deep autoencoder approaches, especially under practical feedback impairments and low-SNR conditions.

Searching arXiv for InvCSINet and closely related CSI-feedback papers. InvCSINet is an information-preserving channel state information (CSI) feedback framework for frequency division duplex (FDD) massive MIMO that replaces the usual non-invertible deep autoencoder compression pipeline with an invertible neural network (INN), and augments that INN with endogenous quantization, differentiable bit-channel distortion modeling, and latent-domain information compensation (Tian et al., 27 Jul 2025). In the formulation reported for InvCSINet, CSI feedback is treated not merely as dimensionality reduction but as a jointly optimized compression–transmission–reconstruction process in which quantization and channel impairments are explicitly integrated into training. The framework is presented for OFDM FDD massive MIMO systems and is evaluated on DeepMIMO under practical assumptions including BPSK modulation and AWGN feedback channels, with normalized mean square error (NMSE) as the primary performance metric (Tian et al., 27 Jul 2025).

1. Problem setting and motivation

InvCSINet is defined in the context of CSI feedback for OFDM FDD massive MIMO systems with $N_c$ subcarriers, $N_t \gg 1$ base-station antennas, and $N_r$ user-equipment antennas. At subcarrier $n$, the downlink CSI is represented as

$\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$

and the full-band CSI is stacked as

$$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$

with total real-valued dimension

$N = 2N_r N_t N_c.$

The feedback pipeline is formulated through UE-side compression, quantization, bit-channel distortion, and BS-side reconstruction: $$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$ followed by $Q(\mathbf{z})$, then

$\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$

and finally

$N_t \gg 1$0

under the objective

$N_t \gg 1$1

(Tian et al., 27 Jul 2025).

The motivation for InvCSINet is a critique of standard deep autoencoder-based CSI feedback. In the reported account, conventional deep learning-based methods are described as ill-posed in two senses: compression is irreversible, and quantization as well as channel noise are typically ignored during training (Tian et al., 27 Jul 2025). The proposed framework responds by replacing irreversible encoder–decoder compression with a bijective latent transform and by embedding practical feedback impairments directly into the trainable system.

A plausible implication is that InvCSINet belongs to a broader line of work in which invertibility is used to avoid representational bottlenecks introduced by lossy latent mappings. Related 2025 work on invertible image compression likewise attributes high-rate performance gains to the absence of an information bottleneck in the transform itself (Tu et al., 27 Mar 2025). In InvCSINet, however, that principle is specialized to CSI feedback and coupled to channel-aware differentiable modules rather than image entropy coding.

2. Invertible reformulation of CSI feedback

The central architectural premise of InvCSINet is that an INN can be used as a bijective transform between the CSI tensor and a structured latent representation. The framework implements

$N_t \gg 1$2

with inverse mapping

$N_t \gg 1$3

where $N_t \gg 1$4 is the transmitted compressed feature and $N_t \gg 1$5 is an auxiliary feature capturing the part of $N_t \gg 1$6 not carried by $N_t \gg 1$7 (Tian et al., 27 Jul 2025).

The framework relies on the distribution-preservation property cited for INNs: if $N_t \gg 1$8 is bijective and maps $N_t \gg 1$9 to $N_r$0, then $N_r$1 maps $N_r$2 back to $N_r$3 (Tian et al., 27 Jul 2025). Within InvCSINet, this gives a theoretical basis for describing the transform as information-preserving, provided the full latent state $N_r$4 is retained. Since only $N_r$5 is actually fed back in the CSI setting, the BS reconstructs using a sampled auxiliary variable $N_r$6, rather than a deterministically transmitted complement.

This design differs from invertible formulations in other domains. For example, in reversible image conversion, recent work has argued that augmenting an underdetermined map with random latent variables does not resolve ill-posedness and instead proposes a well-posed approximate left inverse via an overdetermined system with non-zero Gram determinant (Huang et al., 4 Aug 2025). InvCSINet does not adopt that well-posed reformulation; it instead keeps the conventional INN latent-splitting pattern and learns the reconstruction process under distribution matching, quantization, and channel distortion (Tian et al., 27 Jul 2025). This suggests that “information-preserving” in InvCSINet refers to the bijective transform prior to transmission impairments, rather than to exact deterministic recoverability from the transmitted feedback alone.

3. Architecture and signal flow

The complete InvCSINet pipeline begins by transforming CSI to the angular domain using 2D DFT, concatenating real and imaginary parts, and splitting the resulting tensor into two parts, $N_r$7 and $N_r$8 (Tian et al., 27 Jul 2025). This segmentation is performed according to the target compression ratio so that $N_r$9 matches the dimension of the compressed latent $n$0 and $n$1 matches the auxiliary dimension $n$2. The complex CSI is reshaped into patches of size $n$3, then split along the channel dimension (Tian et al., 27 Jul 2025).

The overall data flow is

$n$4

then

$n$5

followed at the BS by

$n$6

(Tian et al., 27 Jul 2025).

Each invertible block contains two affine coupling layers. For the $n$7-th block,

$n$8

$n$9

with inverse

$\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$0

$\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$1

(Tian et al., 27 Jul 2025). The functions $\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$2 are generic CNN or FCNN modules; in simulation, $\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$3 is set to the identity. Permutation layers are inserted between blocks to improve feature mixing (Tian et al., 27 Jul 2025). The network is trained bidirectionally, alternating forward and backward passes and accumulating gradients from both directions.

The architecture uses 3 invertible blocks in the reported implementation (Tian et al., 27 Jul 2025). The paper also claims a model-complexity advantage over deep autoencoder baselines because encoder and decoder share the same invertible parameters, rather than being realized as separate networks (Tian et al., 27 Jul 2025).

4. Quantization, channel distortion, and information compensation

A distinguishing feature of InvCSINet is that practical feedback impairments are integrated endogenously rather than appended as external post-processing. The architecture inserts three modules after the INN latent $\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$4: Differentiable Adaptive Quantization (DAQ), Differentiable Bit-Channel Distortion (D-BCD), and Information Compensation (IC) (Tian et al., 27 Jul 2025).

DAQ

DAQ quantizes each feature dimension of $\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$5 into $\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$6 bits, with

$\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$7

quantization levels. The quantized vector is

$\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$8

with learnable quantization points $\mathbf{H}_n \in \mathbb{C}^{N_r \times N_t},$9 (Tian et al., 27 Jul 2025). For ideal uniform quantization, the $$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$0-th component resolution is

$$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$1

but the paper emphasizes that ideal quantization depends on the discontinuous sign function

$$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$2

and is therefore non-differentiable (Tian et al., 27 Jul 2025).

To obtain differentiability, InvCSINet uses the soft approximation

$$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$3

with sharpness parameter $$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$4. The adaptive non-uniform quantizer is then written

$$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$5

with learnable parameters

$$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$6

subject to

$$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$7

(Tian et al., 27 Jul 2025). The parameters $$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$8, $$\mathbf{H} = [\mathbf{H}_1,\dots,\mathbf{H}_{N_c}],$$9, and $N = 2N_r N_t N_c.$0 are interpreted respectively as learned decision boundaries, step sizes between quantization levels, and base offset. Quantization points are defined recursively by

$N = 2N_r N_t N_c.$1

$N = 2N_r N_t N_c.$2

(Tian et al., 27 Jul 2025).

D-BCD

After quantization, the symbol is subjected to bit errors through D-BCD, which is designed to mimic the entire bitstream transmission chain in differentiable form. For one feature dimension,

$N = 2N_r N_t N_c.$3

where $N = 2N_r N_t N_c.$4 (Tian et al., 27 Jul 2025).

A soft assignment vector $N = 2N_r N_t N_c.$5 is first defined as

$N = 2N_r N_t N_c.$6

with sharpness parameter $N = 2N_r N_t N_c.$7 (Tian et al., 27 Jul 2025). Channel effects are then modeled by a transition probability matrix $N = 2N_r N_t N_c.$8, where

$N = 2N_r N_t N_c.$9

For BPSK over AWGN with SNR $$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$0, the reported form is

$$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$1

where $$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$2 is the Hamming distance between binary labels of the two quantization symbols (Tian et al., 27 Jul 2025). The detection distribution is

$$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$3

Because direct sampling from $$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$4 is non-differentiable, InvCSINet uses Gumbel-Softmax reparameterization: $$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$5 where $$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$6, $$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$7 are i.i.d. Gumbel noises, and $$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$8 is temperature. The relaxed detected feature is then

$$\mathbf{z} = f(\mathbf{H};\theta_f), \quad \mathbf{z}\in\mathbb{R}^M,\; M\ll N,$$9

(Tian et al., 27 Jul 2025). During testing, the reported implementation uses standard digital modulation and ML detection rather than the relaxed training surrogate.

IC

The IC module addresses the distortion of $Q(\mathbf{z})$0 by reconstructing the latent feature

$Q(\mathbf{z})$1

where $Q(\mathbf{z})$2 is a LAN-like denoising network consisting of a residual connection and a CNN (Tian et al., 27 Jul 2025). Its goal is to make $Q(\mathbf{z})$3 match the distribution of the original compressed latent $Q(\mathbf{z})$4.

The auxiliary variable is also made learnable. Instead of a fixed standard normal prior, InvCSINet uses

$Q(\mathbf{z})$5

with trainable $Q(\mathbf{z})$6 and $Q(\mathbf{z})$7, and reparameterization

$Q(\mathbf{z})$8

(Tian et al., 27 Jul 2025).

5. Training objective and theoretical basis

InvCSINet is optimized with a joint objective

$Q(\mathbf{z})$9

with $\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$0 (Tian et al., 27 Jul 2025). The reconstruction term is CSI-domain mean square error: $\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$1

The distribution-matching term is based on maximum mean discrepancy (MMD). In ideal INN training, the paper states

$\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$2

whereas in the practical InvCSINet setting this becomes

$\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$3

(Tian et al., 27 Jul 2025). The empirical MMD uses the inverse multiquadric kernel

$\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$4

with joint kernel

$\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$5

(Tian et al., 27 Jul 2025).

The paper reports a theorem that if the MMD loss converges to zero, then the reconstructed CSI distribution matches the true CSI distribution: $\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$6 (Tian et al., 27 Jul 2025). A corresponding theorem is stated for InvCSINet under quantization and channel distortion, asserting that under zero distribution-matching loss, the final reconstructed CSI matches the source distribution. This is a distributional statement rather than a pointwise exact-recovery guarantee.

This theoretical stance marks an important distinction from other invertible paradigms. In well-posed reversible image conversion, exact left-invertibility is associated with an augmented operator satisfying $\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$7 (Huang et al., 4 Aug 2025). InvCSINet instead bases its recovery argument on bijective transformation plus MMD-based distribution matching under practical impairments (Tian et al., 27 Jul 2025). This suggests two different interpretations of invertibility in applied systems: exact structural left-invertibility in one line of work, and distribution-preserving latent transport in another.

6. Evaluation, comparisons, and significance

The primary evaluation metric for InvCSINet is

$\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$8

(Tian et al., 27 Jul 2025). Training also tracks MMD. The reported simulation setup uses the DeepMIMO dataset, with an outdoor scene containing two streets and an intersection, carrier frequency $\hat{\mathbf{z}} = \Omega(Q(\mathbf{z})),$9 GHz, bandwidth $N_t \gg 1$00 MHz, $N_t \gg 1$01 subcarriers, $N_t \gg 1$02 BS antennas, and $N_t \gg 1$03 UE antennas (Tian et al., 27 Jul 2025). The train–test split is 70%/30%, corresponding to 12,670 training samples and 5,430 test samples, with batch size 128, 1000 epochs, Adam, initial learning rate $N_t \gg 1$04, learning-rate decay to 0.9 every 20 epochs, $N_t \gg 1$05, and MMD kernel constant $N_t \gg 1$06 (Tian et al., 27 Jul 2025).

The paper compares InvCSINet against CsiNet, CsiNet+, DNNet, and ATNet (Tian et al., 27 Jul 2025). The comparison is structured around several distinctions. CsiNet is described as deep-autoencoder-based and assumes ideal transmission. CsiNet+ uses $N_t \gg 1$07-law quantization with an offline dequantizer but not full end-to-end training. DNNet addresses channel noise but not quantization. ATNet considers both quantization and channel noise but uses multi-step training and gradient approximations (Tian et al., 27 Jul 2025).

The reported empirical findings are that an INN-based scheme outperforms CsiNet even under ideal transmission, that full InvCSINet outperforms all baselines under practical transmission conditions, and that the advantage is especially pronounced at low SNR because quantization and bit-channel distortion are jointly modeled (Tian et al., 27 Jul 2025). The ablations attribute about 0.1–1.5 dB degradation to removing DAQ, about 0.5–11.8 dB to removing D-BCD, and about 0.004–0.4 dB to removing IC (Tian et al., 27 Jul 2025). On that basis, the paper identifies D-BCD as the most critical component when channel noise dominates, DAQ as important when quantization dominates, and IC as a residual correction mechanism.

The significance claimed for InvCSINet is twofold. First, it reframes deep CSI feedback as an invertible, distribution-matching reconstruction problem rather than a conventional lossy compression problem (Tian et al., 27 Jul 2025). Second, it presents quantization and feedback-channel distortion not as exogenous nuisances but as trainable elements of the end-to-end system. A plausible implication is that InvCSINet is best understood not simply as an INN-based replacement for CsiNet, but as a practical systems formulation in which latent invertibility, quantizer adaptation, channel simulation, and denoising compensation are co-designed.

7. Relation to broader invertible learning literature

InvCSINet sits within a 2025 trend of applying invertible neural architectures to domains traditionally dominated by autoencoders. One related example is a multi-scale invertible neural network for wide-range variable-rate learned image compression, where the transform is likewise treated as bijective and information loss is localized to quantization rather than representation learning (Tu et al., 27 Mar 2025). That work argues that invertible transforms are especially advantageous at high bit rates because the transform itself does not impose an information bottleneck (Tu et al., 27 Mar 2025). InvCSINet adopts a similar philosophical premise for CSI feedback: preserving information through the latent transform improves reconstruction fidelity relative to irreversible autoencoder compression (Tian et al., 27 Jul 2025).

At the same time, InvCSINet differs from more recent work that questions whether latent-variable augmentation truly resolves inverse-problem ill-posedness. In reversible image conversion, the introduction of random latent variables is criticized as a source of uncertainty, and a well-posed invertible $N_t \gg 1$08 convolution is proposed to eliminate reliance on random sampling (Huang et al., 4 Aug 2025). InvCSINet, by contrast, still reconstructs from a transmitted feature $N_t \gg 1$09 together with a sampled auxiliary variable $N_t \gg 1$10 (Tian et al., 27 Jul 2025). This is not a contradiction of the paper’s claims, because InvCSINet’s central guarantee is distributional rather than deterministic. However, it highlights a possible conceptual misunderstanding in the reception of invertible methods: bijectivity of the internal transform does not by itself imply deterministic recoverability from only the transmitted subset of latent variables.

Within its own formulation, InvCSINet’s contribution is therefore specific and delimited. It does not claim exact inversion from quantized and noisy compressed latents alone; instead, it proposes a unified end-to-end mechanism that uses a bijective latent transform, adaptive quantization, differentiable channel distortion, and latent compensation to improve NMSE and robustness under practical CSI feedback impairments (Tian et al., 27 Jul 2025). In that sense, InvCSINet is most precisely characterized as an invertible, impairment-aware CSI feedback architecture for FDD massive MIMO rather than as a general solution to all invertibility or well-posedness issues in compressed sensing or inverse reconstruction.