ConvBiS6 Layer for Point Clouds

• ConvBiS6 is a specialized token mixer for point clouds that combines bidirectional selective state-space (MHS6) and explicit 1D convolution to model global context and local geometric details.
• It leverages a 3D Hilbert-curve-based shuffle serialization to create a 1D token sequence, addressing the limitations of causal S6 and ensuring effective local dependency modeling.
• Empirical evaluations on datasets like ModelNet40 and S3DIS show that the hybrid design of ConvBiS6 yields state-of-the-art performance in point cloud classification and segmentation.

ConvBiS6 is the token-mixing layer introduced in HydraMamba for point cloud learning. It replaces self-attention inside a MetaFormer-style block and is designed for serialized point sequences produced by a 3D Hilbert-curve-based shuffle serialization scheme. Its defining property is the joint use of a bidirectional selective state-space path—implemented in practice with Multi-Head S6 (MHS6)—and an explicit 1D convolution branch, so that local geometries and global context dependencies are modeled within the same layer (Qu et al., 26 Jul 2025).

1. Origin and design rationale

ConvBiS6 is introduced in Section 4.3, “HydraMamba block,” as the core token mixer in HydraMamba, an encoder–decoder point cloud backbone. Each HydraMamba block follows a MetaFormer-style pattern, namely Norm $\rightarrow$ token mixer $\rightarrow$ Norm $\rightarrow$ MLP, with residual connections; the token mixer is ConvBiS6 rather than self-attention (Qu et al., 26 Jul 2025).

The layer is motivated by two limitations identified in earlier S6-based point-cloud methods. First, plain S6 is causal and forward-only, so each token can only access its history. Prior work may add bidirectionality, but the contextual signal is still compressed into a hidden state and does not explicitly model local structure. Second, even bidirectional S6 still “considers global connectivity by compressing all contextual information into the history-hidden state” and therefore “lacks a mechanism for modeling local dependencies between adjacent points.” This deficiency is especially consequential for point clouds, where locality is central to edges, lines, and small-scale geometric structure (Qu et al., 26 Jul 2025).

The name encodes the intended composition. “Conv” denotes a 1D grid convolution branch over the serialized sequence, “Bi” denotes bidirectional S6, and “S6” denotes the selective state-space model used as the sequential operator. In implementation, the S6 operator is replaced by MHS6, but the layer name remains ConvBiS6. This naming reflects function rather than a literal enumeration of internal primitives (Qu et al., 26 Jul 2025).

2. Placement within HydraMamba and architectural composition

ConvBiS6 operates on a serialized point sequence

$$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$

where $B$ is batch size, $L$ is the sequence length after serialization, and $D$ is the feature dimension. Within a HydraMamba block, serialized point features enter ConvBiS6, the mixed output is then passed through the feed-forward MLP, and the overall block preserves the residual, Transformer-style organization (Qu et al., 26 Jul 2025).

Architecturally, ConvBiS6 is a two-branch sequence mixer.

The first branch is a bidirectional S6 path, implemented with MHS6. One scan proceeds from index $1 \rightarrow L$, while a second scan proceeds from $L \rightarrow 1$. The two directions are parallelized so that each token receives a global receptive field over the serialized point cloud. The paper does not spell out the exact fusion formula for forward and backward outputs; a plausible implementation, already suggested in the technical summary, is summation or concatenation followed by a linear projection.

The second branch is a 1D convolution over the sequence dimension. It uses a kernel size $k > 1$ and may be instantiated as either depth-wise Conv1D or standard Conv1D. Both forms were tested in ablation, with negligible difference in overall accuracy, which supports the interpretation that the critical property is explicit locality modeling rather than the specific convolution type (Qu et al., 26 Jul 2025).

The fused output of the bidirectional state-space branch and the convolution branch becomes the mixed representation used by the surrounding MetaFormer block. ConvBiS6 is used in every HydraMamba block across encoder and decoder stages. At the network level, HydraMamba contains an embedding layer, an encoder with transition-down modules, a decoder with transition-up modules, and task heads based on either global average pooling plus MLP for recognition or per-point MLP for segmentation (Qu et al., 26 Jul 2025).

3. Mathematical formulation

HydraMamba reviews the continuous-time state-space model in the form

$\rightarrow$0

with hidden state $\rightarrow$1, input $\rightarrow$2, and parameters $\rightarrow$3, $\rightarrow$4, and $\rightarrow$5. Under zero-order-hold discretization with step size $\rightarrow$6,

$\rightarrow$7

where

$\rightarrow$8

In S6, $\rightarrow$9 and $\rightarrow$0 are input-dependent and are computed with a parallel scan algorithm, giving linear complexity in sequence length (Qu et al., 26 Jul 2025).

ConvBiS6 uses MHS6 rather than a single-head S6. Given

$\rightarrow$1

the input is reshaped into

$\rightarrow$2

where $\rightarrow$3 is the number of heads. For each head $\rightarrow$4, the layer computes head-specific $\rightarrow$5 and $\rightarrow$6 projections,

$\rightarrow$7

$\rightarrow$8

and a per-token step size

$\rightarrow$9

The softplus parameterization ensures $$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$0. Using $$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$1, the layer forms dynamic $$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$2 and $$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$3,

$$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$4

followed by

$$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$5

The final MHS6 output is obtained by concatenating all heads along the channel dimension (Qu et al., 26 Jul 2025).

Bidirectional MHS6 applies this procedure to the original and reversed sequence. The paper does not provide a closed-form fusion equation; a natural formulation, explicitly identified as such in the technical notes, is either

$$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$6

or

$$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$7

The convolution branch is correspondingly simple: $$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$8 An expert-level view of the layer is therefore

$$\boldsymbol{x} \in \mathbb{R}^{B \times L \times D},$$9

followed by normalization, residual connection, and MLP within the enclosing HydraMamba block (Qu et al., 26 Jul 2025).

4. Serialization, locality, and global context

ConvBiS6 relies on HydraMamba’s shuffle serialization. Points are mapped to a 1D sequence with a 3D Hilbert curve, and the six traversal variants $B$0, $B$1, $B$2, $B$3, $B$4, and $B$5 are used as alternative orderings. Rather than fixing one ordering, HydraMamba assigns one variant per block and shuffles these assignments across layers (Qu et al., 26 Jul 2025).

The significance of this choice is geometric. The Hilbert curve is described as strongly locality-preserving: neighboring points in 3D are likely to remain close in the 1D sequence. Consequently, a small 1D convolution kernel over the serialized sequence functions as local aggregation over Hilbert-neighbor points, approximating local 3D neighborhoods without explicit $B$6NN or ball-query construction. Different axis-priority variants induce complementary notions of locality and neighborhood, so the shuffle strategy exposes different blocks to different local geometric views (Qu et al., 26 Jul 2025).

Global context is handled by the bidirectional S6 branch. S6 is a global sequence model with linear complexity, but its native causal form is not sufficient for visual and 3D tasks that require the full sequence context. HydraMamba therefore uses both forward and backward scans, so that each token is influenced by the entire serialized point set. The hidden-state evolution

$B$7

remains the core mechanism, but with input-dependent transitions and opposite scan directions. MHS6 adds multiple parallel subspaces for learning distinct geometric dependency patterns (Qu et al., 26 Jul 2025).

This combination addresses both limitations identified at the outset. Shuffle serialization mitigates bias induced by any single traversal order, while the explicit convolution branch prevents locality from being left entirely to compression in the state-space hidden state. The resulting design is analogous, in conceptual terms, to local-attention-plus-global-attention hybrids, but here the decomposition is realized as SSM plus 1D convolution rather than attention plus attention (Qu et al., 26 Jul 2025).

5. Empirical evidence

The clearest evidence for ConvBiS6 comes from the ModelNet40 ablation in Table 6, which isolates bidirectionality and convolutional locality modeling (Qu et al., 26 Jul 2025).

Configuration	OA	Brief implication
Vanilla	89.13%	No bidirectionality, no convolution
Bidirectional S6 only	92.40%	Global bidirectional modeling is important
Conv-only	89.88%	Local convolution alone is insufficient
BiS6 + depth-wise conv	93.96%	Best reported combination
BiS6 + standard conv	93.95%	Nearly identical to depth-wise conv

The numerical pattern is structurally informative. Bidirectional S6 alone improves OA by $B$8 points over the vanilla baseline, showing that making S6 non-causal in the serialized domain is essential for global modeling. Conv-only improves OA by $B$9 points, indicating that local aggregation does help, but not enough to replace state-space global interaction. The combined design is best, and the difference between depth-wise and standard convolution is negligible, supporting the claim that explicit locality itself is the decisive factor (Qu et al., 26 Jul 2025).

HydraMamba, which uses ConvBiS6 in all blocks, reports state-of-the-art results on multiple tasks. On ModelNet40 classification it achieves $L$0 OA; on ShapeNet part segmentation it achieves $L$1 Ins. mIoU; on S3DIS semantic segmentation it achieves $L$2 mIoU. The paper describes the ShapeNet result as the first time an SSM surpasses Transformer-based paradigms on that task. For S3DIS, the reported $L$3 mIoU slightly surpasses Point Transformer V3 at $L$4 and Pamba at $L$5, while HydraMamba directly processes entire scenes with $L$6K points globally, a use case enabled by the linear-time S6 component of ConvBiS6 (Qu et al., 26 Jul 2025).

Qualitative evidence in the appendix is also consistent with the layer’s intended role: the visualizations reportedly show improved boundary segmentation and part delineation relative to PointMamba and Point Transformer. This suggests that the local branch contributes to sharper geometric discrimination while the bidirectional state-space branch preserves scene-scale context.

6. Implementation characteristics and operating regime

ConvBiS6 preserves feature dimensionality: both input and output are of shape $L$7. The exact convolution kernel size is not explicitly tabulated, and the paper does not provide a dedicated hyperparameter table for ConvBiS6. What is explicit is that the layer uses a Conv1D kernel with $L$8, and that both depth-wise and standard variants were tested (Qu et al., 26 Jul 2025).

For MHS6, the default initial head count is $L$9 heads for the initial embedding dimension. The head count scales proportionally in deeper stages with larger feature dimensions, although exact per-stage values are not enumerated in the main text. The ablation over head count reports the following OA values on ModelNet40: single S6 without multi-head design, $D$0; $D$1 heads, $D$2; $D$3 heads, $D$4; $D$5 heads, $D$6; and $D$7 heads, $D$8. The paper’s practical reading is that there is a sweet spot: too few heads underuse representational capacity, whereas too many degrade subspace quality (Qu et al., 26 Jul 2025).

Normalization and feed-forward processing follow the MetaFormer pattern, so ConvBiS6 is wrapped by layer normalization and an MLP with nonlinearity inside each HydraMamba block. The exact number of HydraMamba blocks per stage is not enumerated in the main text, but every block contains one ConvBiS6 layer. A plausible implication is that ConvBiS6 should be understood as the recurrent structural unit of the backbone rather than as an auxiliary module (Qu et al., 26 Jul 2025).

At the system level, HydraMamba’s efficiency claims are relevant because ConvBiS6 is its central token mixer. The S6 path uses the parallel scan algorithm and GPU-friendly kernels inspired by FlashAttention, giving linear memory and time in sequence length. On S3DIS, the paper reports $D$9 GB memory per scene for HydraMamba versus $1 \rightarrow L$0 GB for Point Transformer V3, and latency of $1 \rightarrow L$1 ms per scene, which is faster than some attention baselines and slightly slower than Point Transformer V3 at $1 \rightarrow L$2 ms (Qu et al., 26 Jul 2025).

7. Nomenclature, scope, and related interpretations

In current arXiv usage, ConvBiS6 is specifically the HydraMamba layer described above. The term can nevertheless be misread because its components—“Conv,” “Bi,” and “S6”—are semantically suggestive outside that context. The record in the cited literature is explicit on this point for at least two cases: the name does not appear in “Design of Efficient Convolutional Layers using Single Intra-channel Convolution, Topological Subdivisioning and Spatial ‘Bottleneck’ Structure” (Wang et al., 2016), and it does not appear in “Blended Convolution and Synthesis for Efficient Discrimination of 3D Shapes” (Ramasinghe et al., 2019).

That said, the data support several interpretive associations. One reading connects the term to efficient convolutional decomposition: single intra-channel convolution, topological subdivisioning, and spatial bottlenecks together form an efficient replacement for dense convolution, and the phrase “ConvBiS6 layer” can be read as a natural but non-canonical label for such a block (Wang et al., 2016). A second reading connects it to the “Blended Convolution and Synthesis” layer, where projection to a latent representation in the unit ball $1 \rightarrow L$3 is fused with a novel 3D convolution operator; the technical notes explicitly describe this as a ConvBiS-style 3D convolution, though not with HydraMamba’s meaning (Ramasinghe et al., 2019). A third, more conjectural connection associates the name with hybrid cosine-basis convolutions and treats “ConvBiS6” as a hypothetical specialized convolutional layer built from analytic filter parameterizations (Ciurana et al., 2019).

These alternative readings are not definitions of ConvBiS6 in the HydraMamba sense. A plausible implication is that acronymic similarity obscures three distinct design lineages: efficient depthwise-plus-projection convolutional factorization, spectral blended convolution and synthesis in $1 \rightarrow L$4, and cosine-parameterized hybrid convolutions. In contrast, the established technical meaning of ConvBiS6 is precise: a two-branch token mixer for Hilbert-serialized point clouds that combines bidirectional MHS6 for global dependency modeling with Conv1D for explicit locality learning (Qu et al., 26 Jul 2025).