# TeraPool-SDR: An 1.89TOPS 1024 RV-Cores 4MiB Shared-L1 Cluster for Next-
Generation Open-Source Software-Defined Radios

Yichao Zhang<EMAIL_ADDRESS>0009-0008-7508-599X Integrated Systems
LaboratoryETH ZürichZürichSwitzerland ,  Marco Bertuletti
<EMAIL_ADDRESS>0000-0001-7576-0803 Integrated Systems
LaboratoryETH ZürichZürichSwitzerland ,  Samuel Riedel<EMAIL_ADDRESS>0000-0002-5772-6377 Integrated Systems LaboratoryETH ZürichZürichSwitzerland ,
Matheus Cavalcante<EMAIL_ADDRESS>0000-0001-9199-1708 Integrated
Systems LaboratoryETH ZürichZürichSwitzerland ,  Alessandro Vanelli-Coralli
<EMAIL_ADDRESS>0000-0002-4475-5718 Integrated Systems LaboratoryETH
ZürichZürichSwitzerland Università di BolognaBolognaItaly and  Luca Benini
<EMAIL_ADDRESS>0000-0001-8068-3806 Integrated Systems LaboratoryETH
ZürichZürichSwitzerland Università di BolognaBolognaItaly

(2024)

###### Abstract.

Radio Access Networks (RAN) workloads are rapidly scaling up in data
processing intensity and throughput as the 5G (and beyond) standards grow in
number of antennas and sub-carriers. Offering flexible Processing Elements
(PEs), efficient memory access, and a productive parallel programming model,
many-core clusters are a well-matched architecture for next-generation
software-defined RANs, but staggering performance requirements demand a high
number of PEs coupled with extreme Power, Performance and Area (PPA)
efficiency. We present the architecture, design, and full physical
implementation of Terapool-SDR, a cluster for Software Defined Radio (SDR)
with 1024 latency-tolerant, compact RV32 PEs, sharing a global view of a 4 Mi
, $4096$-banked, L1 memory. We report various feasible configurations of
TeraPool-SDR featuring an ultra-high bandwidth PE-to-L1-memory interconnect,
clocked at 730 M , 880 M , and 924 M
(TT/$0.80\text{\,}\mathrm{V}$/$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$) in
$12\text{\,}\mathrm{nm}$ FinFET technology. The TeraPool-SDR cluster achieves
high energy efficiency on all SDR key kernels for 5G RANs: Fast Fourier
Transform (93 G /), Matrix-Multiplication (125 G /), Channel Estimation (96 G
/), and Linear System Inversion (61 G /). For all the kernels, it consumes
less than 10, in compliance with industry standards.

Many-core, RISC-V, Software-Defined Radios, Physical Design

††submissionid: 6003††journalyear: 2024††copyright: acmlicensed††conference:
Great Lakes Symposium on VLSI 2024; June 12–14, 2024; Clearwater, FL,
USA††booktitle: Great Lakes Symposium on VLSI 2024 (GLSVLSI ’24), June 12–14,
2024, Clearwater, FL, USA††doi: 10.1145/3649476.3658735††isbn:
979-8-4007-0605-9/24/06††ccs: Computer systems organization Multicore
architectures††ccs: Hardware Digital signal processing††ccs: Networks Topology
analysis and generation

##  1\. Introduction

Implementing the 5th-Generation (5G) Radio Access Networks (RAN) standards and
beyond, requires flexible high-performance computing systems that can sustain
large computational workloads and memory footprints in a tight power envelope.
To guarantee time-to-market and performance, industry is moving from
architectures where network functions are offloaded to specialized
accelerators to an open and disaggregated Software Defined Radio (SDR)
paradigm (Mitola, 1995), building processing pipelines as a chain of
programmable components. Off-the-shelf products include programmable (EdgeQ,
2023; Qualcomm, 2023; Marvell, 2023) or reconfigurable (Xilinx, 2023) hardware
with their in-house software libraries. Their designs target
$10\text{\,}\mathrm{G}\mathrm{b}\mathrm{p}\mathrm{s}$ uplink bandwidth at less
than $50\text{\,}\mathrm{W}$ power consumption (EdgeQ, 2023; Qualcomm, 2023),
on Fast Fourier Transform (FFT) for Orthogonal Frequency Division Multiplexing
(OFDM), Matrix Multiplication (MatMul) for Beam Forming (BF), Channel
Estimation (CHE), and Linear System Inversion (SysInv) for Multiple-Input,
Multiple-Output (MIMO) detection: the big heterogeneous workloads of the 7.X
Open Radio Access Networks (O-RAN) functional splits (Larsen et al., 2019).
However, closed proprietary does not allow open research, community-based
hardware-software co-design, a propulsive force for SDRs.

Focusing on programmable solutions, the many-core cluster offers energy-
efficient parallelism. A successful architectural pattern is shared-L1 cluster
that eliminates the hardware and software overheads incurred by replicating a
private-L1-base many-core cluster for performance purposes (Ditzel et al.,
2021; Ginosar et al., 2021) (synchronization, inter-cluster data allocation-
splitting, and workload distribution among clusters). Increasing the shared
memory many-core cluster scale is also highly desirable to exploit the
embarrassingly parallel features of SDR processing. Focusing on a concrete
example, the parallelization of 5G Physical Uplink Shared Channel (PUSCH)
receiver processing (Bertuletti et al., 2023) exhibits strong data
dependencies: for instance, different FFT streams of the OFDM stage need to be
merged and multiplied by a matrix of coefficients in the BF stage. If a single
cluster shared-L1 is not large enough, MatMuls can be tiled, but the OFDM data
must first be merged through the upper levels of the memory hierarchy. Table 1
resumes this data transfer overhead for each stream, comparing four loosely-
coupled $1\text{\,}\mathrm{MiB}$ clusters and a single shared-L1
$4\text{\,}\mathrm{MiB}$ cluster, working on $64$ OFDM-antennas of
$3276$-subcarriers and a 32x64x3276 BF MatMul. In the first case, the
$32\text{\,}\mathrm{b}$ words $64$ OFDM-antennas are divided into groups of
$16$, the output is transferred to L2 and the MatMul is tiled over rows for
each processing streams. In the second, the $64$ antennas are all processed in
L1 and the MatMul stage only requires the transfer of coefficients.

Table 1. The data transfer for 5G PUSCH in different clusters.

|  $1\text{\,}\mathrm{MiB}$ Clusters |  $4\text{\,}\mathrm{MiB}$ Cluster
---|---|---
#Clusters$\times$Cores | 4$\times$256 | 1$\times$1024
(OFDM-antennas$\times$subcarriers)/Cluster |  16x3276 |  64x3276
BF(MatMul) Size/Cluster |  32x64x819 |  32x64x3276
Max. L1-occupation ($\mathrm{KiB}$) | 520 | 2055
Transfer-out OFDM-antennas ($\mathrm{KiB}$) | 205 | -
Transfer-in BF-inputs ($\mathrm{KiB}$) | 213 | 8
Total Transfer Overhead ($\mathrm{KiB}$) | 418 | 8

The existing many-core cluster only up to tens complex processors with shared
L1 space: Kalray’s MPPA-256 features a 16-cores shared
$2\text{\,}\mathrm{MiB}$ cluster-based architecture, Esperanto ET-SoC-1’s
$4\text{\,}\mathrm{MiB}$ L1 are shared by $32$ vector cores and Ramon RC64 has
$64$ cores with $4\text{\,}\mathrm{MiB}$ shared memory (Kamaleldin et al.,
2020; Ditzel et al., 2021; Ginosar et al., 2021), which are not scalable to
the parallel processing of 5G O-RAN large problems. The OFDM and BF are indeed
very compute-intensive steps. With $N_{SC}$ sub-carriers, $N_{R}$ antennas,
$N_{B}$ beams, the computational complexity of the kernels is $O(N_{R}\times
N_{SC}log(N_{SC})+N_{R}\times N_{SC}\times N_{B})$ real Multiply Accumulate
(MAC) operations. In PUSCH, up to $14$ FFT & MatMul streams are executed per
$1$ms Transition Time Interval (TTI), which in a typical use-case
($N_{SC}=3276$, $N_{B}=32$), takes $\sim$$0.8\text{\,}\mathrm{TOPS}$ with $64$
receivers (5G) and $\sim$$1.8\text{\,}\mathrm{TOPS}$ with $128$ (6G)
receivers. This requires a scalable cluster with thousands of programmable
agile cores, running at near GHz frequency.

This work introduces TeraPool-SDR, a peak $1.89\text{\,}\mathrm{TOPS}$
cluster, featuring $1024$ RISC-V cores with hierarchical low-latency
interconnections to a fully shared $4\text{\,}\mathrm{MiB}$, $4096$-banked L1
Scratchpad Memory (SPM). The contributions are:

  * •

A physical-aware architecture design for O-RAN workloads, based on a three-
level physical implementation hierarchy: the Tile, the SubGroup, and the
Group;

  * •

A detail latency-throughput evaluation of on-chip Non-Uniform Memory Access
(NUMA) latency interconnection;

  * •

A complete physical implementation and Power, Performance and Area (PPA)
analysis of each design configuration, using the cutting-edge
$12\text{\,}\mathrm{nm}$ FinFET technology node;

  * •

A comprehensive performance and energy efficiency evaluation on key 5G-SDR
kernels, showing $0.18\text{\,}\mathrm{-}$$0.84\text{\,}\mathrm{TOPS}$ and
$60\text{\,}\mathrm{-}$$125\text{\,}\mathrm{GOPS}\text{/}\mathrm{W}$ in real
5G 7.X split benchmarks, meets the O-RAN specifications and is the first open-
sourced111https://github.com/pulp-platform/mempool programmable solution for
complete physical layer processing.

##  2\. Architecture

While it was demonstrated that shared-L1 clusters with up to $256$ cores can
be built (Riedel et al., 2023), their performance is still insufficient for
low-latency TTI 5G pipelines. To meet real-life O-RAN requirements, TeraPool-
SDR needs to ”enter into uncharted territories” in terms of core count, aiming
at a 4x increase over the largest cluster reported in the literature. We must
aggressively leverage physical design awareness, building the cluster
hierarchically to ensure high Processing Elements (PEs)-L1 bandwidth and
energy efficiency.

###  2.1. Snitch Core, Tile and Interconnection

TeraPool-SDR’s PEs are single-stage 32-bit RISC-V _Snitch_ cores (Zaruba et
al., 2021) featuring an accelerator port to offload complex instructions to
pipelined functional units. Our Snitch supports the RV32IMAXpulp- img222The
Xpulpimg extension includes domain-specific instructions (Riedel et al.,
2023), e.g., MAC and load-post-increment. Instruction Set Architecture (ISA)
and multiple outstanding transactions to tolerate multi-cycle memory access
latencies. Snitch retires loads out-of-order, yet delivers in-order data to
the execution units.

The basic building block of TeraPool-SDR is the _Tile_ , Figure 1. It contains
$8$ Snitch PEs, with a $32$ instructions private L0 Instruction Cache (I$)
each, and a shared $4\text{\,}\mathrm{K}\mathrm{i}\mathrm{B}$ two-way set-
associative L1 I$. The PEs are tightly coupled to $32$
$1\text{\,}\mathrm{KiB}$ data Static Random-Access Memory (SRAM) L1 banks. The
Advanced eXtensible Interface (AXI) master port, used for L1 I$ refill, and
Direct Memory Access (DMA) transfers, for data-intensive applications, is
shared between all the cores.

A hierarchical topology is essential for an implementable interconnection
between $1024$ PEs and $4096$ memory banks. To keep low-latency Tightly
Coupled Data Memory (TCDM) access, we implement Fully-Connected (FC)
logarithmic crossbars and arbitrators within each design level to have purely
combinational routing. Pipeline registers are optionally added at hierarchy
boundaries to reduce critical paths in the physical design at the cost of
increased latency. Separate request (address, data write, and control) and
response (request ID, data read, and acknowledgment) networks then handle NUMA
accesses.

Figure 1. The TeraPool-SDR Tile architecture, the FC crossbar interconnections
protocol specified in Section 2.1.

The TeraPool-SDR Tile architecture figure shows the fully connected crossbar
interconnection protocol of local and remote Tile request and response

At the Tile level, PEs are connected to the local portion of the cluster’s
shared SPM with a FC crossbar, providing single-cycle zero-load access
latency. Each Tile has a parameterizable number of remote ports to route
requests to other Tiles’ SPM banks via the _remote request interconnect_.
Corresponding responses are routed to the core on the _remote response
interconnect_. To access the SRAM banks, incoming requests directly connect to
the Tile FC local crossbar. The local interconnect provides a master port for
each of the K incoming request ports, resulting in an
$($8$+\text{K})\times$32$$ crossbar.

###  2.2. Cluster Configurations

The TeraPool-SDR Cluster consists of $128$ Tiles, including $1024$ Snitch
cores and $4096$ $1\text{\,}\mathrm{KiB}$ SRAM banks of L1 SPM. Choosing the
hierarchies and the placement of the interconnects within, strongly conditions
the design feasibility. For example, a single upper hierarchy with $128$ Tiles
connected by a $128$$\times$$128$ crossbar is clearly unfeasible. Grouping
$16$ Tiles results in $\text{K}=8$ ports per Tile (one for each Tile-Group)
and delivers high inter-Tile interconnect bandwidth. However, it requires
eight $16$$\times$$16$ FC crossbars within the hierarchy level, and leads to
unmanageable routing congestion. Moreover, it is impossible to place the 8
Tile-Groups in a grid that results in balanced, short access paths and the
Electronic Design Automation (EDA) runtime for the Tile-Group design iteration
is unmanageable. More details on physical feasibility are shown in Section 3.

To make such a huge cluster physically feasible, we propose the following
design strategies for the TeraPool-SDR architecture:

  1. (1)

The topmost hierarchy, the Group, is replicated $4$ times and arranged in a
$2$$\times$$2$ grid to both shorten and balance the diagonal access paths
between the Groups.

  2. (2)

To keep the EDA tool’s runtime manageable, a Group is divided into multiple
finely-tuned physical implementation hierarchies. That is, each Group consists
of $4$ SubGroups, and each SubGroup contains $8$ Tiles.

  3. (3)

Targeting high inter-tile bandwidth, we forward Tiles’ remote Group requests
directly to the Group level, where we implement $32$$\times$$32$ FC crossbars
for each target Group.

Given the interconnect required to establish local connections among Tiles in
a SubGroup, we partition the local interconnection among the four SubGroups,
equipping each with four $8$$\times$$8$ FC crossbars, improving physical
routing feasibility. A Tile’s remote request is routed based on the targeted
hierarchy: local SubGroup, remote SubGroups, or remote Groups, resulting in
$\text{K}=7$ ports: $1$ connects locally to other Tiles in the same SubGroup
via a $8$$\times$$8$ FC crossbar, $3$ connect to Tiles in the other $3$
SubGroups within the local Group via three $8$$\times$$8$ FC crossbars, and
$3$ connect to Tiles in the other $3$ remote Groups via three $32$$\times$$32$
crossbars.

The hierarchy levels provide flexibility to place pipeline registers and break
long remote access paths. Within the Group, registers are placed at each
hierarchy boundary on master ports. The latency between Groups is a hardware-
parameter that trades off the target operating frequency and latency. We call
those parametrizations $\text{TeraPool-SDR}_{\text{1-3-5-X}}$ architecture,
where the subscripts indicate the zero-load cycle latency for core access at
each hierarchy level, i.e., Tile, SubGroup, Group, Cluster. The
$\text{TeraPool-SDR}_{\text{1-3-5-5}}$ design has no extra registers between
the two Groups. A register is added on both request and response paths between
Groups at the Cluster level in the $\text{TeraPool-SDR}_{\text{1-3-5-7}}$
design, increasing the round-trip latency by $2$ cycles. In $\text{TeraPool-
SDR}_{\text{1-3-5-9}}$ and $\text{TeraPool-SDR}_{\text{1-3-5-11}}$, additional
spill registers are added respectively to slave or both master-slave ports of
the Group hierarchy level. The maximum access latency increases by $4$ and $6$
cycles. As an example, the full architecture overview of $\text{TeraPool-
SDR}_{\text{1-3-5-7}}$ is shown in Figure 2. These configurations create a
trade-off between achievable frequency and L1 worst-case latency. A cluster-
level AXI interface is available for connection with external peripherals,
chosen depending on the system design, which is not discussed in this paper.

Figure 2. Bottom-up architecture of $\text{TeraPool-SDR}_{\text{1-3-5-7}}$,
with the interconnection protocol specified in Section 2.1.

Bottom-up architecture of TeraPool-SDR 1-3-5-7 configuration, with the
interconnection protocol

###  2.3. Interconnection latency-throughput trade-off

To compare the performance of various topologies, we replace the cores with
traffic generators producing memory requests following a Poisson process with
a rate $\lambda$ and targeting random and uniformly distributed destination
banks. We present the latency and throughput as a function of the injected
load, measured in requests per core per cycle.

Figure 3. Throughput and average round-trip latency of TeraPool-SDR’s L1
interconnect as a function of the load.

Throughput and average round-trip latency of TeraPool-SDR’s L1 interconnect as
a function of the load

As shown in Figure 3 left, the throughput exhibits a linear-increasing trend
until saturation, as the injected requests start conflicting in the
interconnect to the L1 banks. Increasing the number of pipeline registers
creates a deeper interconnection that accepts more requests before generating
conflicts. Therefore, the saturation load ranges
$0.23$-$0.245\text{\,}\mathrm{request}\text{/}\mathrm{core}\text{/}\mathrm{cycle}$.
Figure 3 right shows the average round-trip latency of a memory request. For a
low injected load, the simulated latency asymptotically approaches the zero-
load access latency, ranging from $4.9$ to $9.3$, depending on the
configuration. As the injected load increases, the average latency grows, its
asymptotic knee going from $10$ to $14$ ($40$% variation) and corresponds to
throughput saturation, at the interconnect congestion. This analysis shows a
clear trade-off between latency and throughput of different configurations.
However, random loads generate an adverse access pattern with no locality
within Tiles. We analyze performance on key O-RAN application kernels in
section 4.

##  3\. Physical Implementation

We implement TeraPool-SDR with GlobalFoundries’ $12\text{\,}\mathrm{nm}$
LPPLUS FinFET technology, using Synopsys’ Fusion Compiler 2022.03 for
synthesis, Place and Route (PnR), determining the power consumption by
Synopsys’ PrimeTime 2022.03 under typical operating conditions
(TT/$0.80\text{\,}\mathrm{V}$/$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$) with
switching activities obtained from post-PnR gate-level simulations with back-
annotated parasitic information. We use a bottom-up implementation flow,
creating abstract design views during assembly, including only interface logic
and register timing information, to reduce the EDA runtime.

###  3.1. Floorplan and Feasibility

We show TeraPool-SDR physical design full view in Figure 4. To fully leverage
the available Back-End-of-Line (BEOL) resources and ease interconnect routing
through Tiles without manual port placement, we flatten the Tile in the
SubGroup, obtaining a $1.52mm\times 1.52mm$ area with a satisfactory
utilization of $55$%, as the first implementation hierarchy level. The SPM
macros of each Tile are grouped and arranged in a U-shape to enclose the local
crossbar, minimizing overall distance and avoiding excessive stacking of
macros. SubGroup and Group blocks are arranged in a point-symmetric grid to
balance the diagonal access paths, with channels in between to place and route
the interconnects. To further improve area utilization, we place interface
ports behind the SubGroup blocks and shrink the channel width until BEOL
resources become limited.

The runtime of EDA tools serves as a fundamental indicator of both the design
optimization effort and viability (Zhou et al., 2017). As mentioned in Section
2.2, although the Cluster configuration comprising $8$ Groups with $16$ Tiles
each delivers high inter-Tile interconnect bandwidth, it is not physically
feasible. The trail implementation shows total EDA runtime of this
configuration is nearly $3.5\times$ that of $\text{TeraPool-
SDR}_{\text{1-3-5-9}}$, with timing optimization accounting for more than
$80$% of the effort, and the routing stage is $5.5\times$ slower than the
other configurations. Despite these high design optimization efforts, the
eight $16\times 16$ interconnects in this Group design generate significant
routing congestion and numerous metal shorts in the post-routing design phase.
Furthermore, routing detours considerably increase the length of timing paths,
making it impossible to close the design with a $500\text{\,}\mathrm{MHz}$
target frequency, under typical operating
conditions(TT/$0.80\text{\,}\mathrm{V}$/$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$).
All TeraPool-SDR configurations are physically feasible.

Figure 4. Placed-and-routed layout annotated view of each TeraPool-SDR
hierarchical instance.

Placed-and-routed layout annotated view of each TeraPool-SDR hierarchical
instance

###  3.2. Peak-Performance & Area

Table 2 presents the post-PnR peak-performance & area results for the most
promising TeraPool-SDR configurations, in comparison to a smaller-scale,
256-core MemPool design in the same technology node. For fair comparison, we
keep $55\text{\,}\mathrm{\%}{}$ area utilization on the base hierarchy of all
the implementations. TeraPool-SDR organizes wisely hierarchical interconnects
that contribute minimal area overhead, accounting for only $8.5$% of the total
design area. TeraPool-SDR breaks the $\mathrm{TOPS}$ wall, achieving peak
performance comprised between $1.50\text{\,}\mathrm{TOPS}$ and
$1.89\text{\,}\mathrm{TOPS}$, depending on the latency configuration. Although
a large-scale physical design leads to longer distance paths, TeraPool-SDR’s
low-latency scheme still keeps a small Fan-Out-of-4 (FO4) delay. When the
configured latency exceeds $11$ cycles, the operating frequency is no longer
constrained by the remote Group access path but by the Snitch core: the
critical path, consisting of only $63$ logic levels, starts at a register
after the instruction cache, passes through Snitch and a request
interconnection, and arrives at the clock gating of an SRAM bank. On the top
hierarchy, TeraPool-SDR loses area efficiency due to the large $32\times 32$
interconnection modules in each Group, requiring more routing area, and the
difficult routing of feedthrough connections across Groups, caused by the
blocking SubGroups placement. These challenges impose physical constraints on
the further scaling of shared-L1-memory architectures beyond the 1000-cores
milepost. We nevertheless believe that exploring 3D-IC solutions will be
beneficial for future research (Kang, 2023).

Table 2. Post-PnR peak-performance & area results.

|  | MemPool256* | TeraPool-SDR1024*
---|---|---|---
Hier. Access Latency [$\mathrm{cycle}$] | 1-3-5 | 1-3-5-7 | 1-3-5-9 | 1-3-5-11
Area [${\mathrm{mm}}^{2}$] | $10.0$ | $68.9$ | $68.9$ | $68.9$
Area Per Core [$\mathrm{m}\mathrm{m}\mathrm{{}^{2}}\mathrm{/}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}$] | $0.039$ | $0.067$ | $0.067$ | $0.067$
Logic Gate [$\mathrm{M}\mathrm{G}\mathrm{E}$] | $44$ | $176$ | $176$ | $176$
Logic Gate Per Core [$\mathrm{M}\mathrm{G}\mathrm{E}\mathrm{/}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}$] | $0.17$ | $0.17$ | $0.17$ | $0.17$
TCDM Spill Register in Group | ✓ | ✓ |  ✓✓ |  ✓✓✓
TCDM Spill Register in Cluster | - | ✓ | ✓ | ✓
Operating Frequency _(Worst)_ [$\mathrm{MHz}$] | $728$ | $530$ | $637$ | $740$
Operating Frequency _(Typ.)_ [$\mathrm{MHz}$] | $915$ | $730$ | $880$ | $924$
Logic Delay [$\mathrm{F}\mathrm{O}\mathrm{4}$] | $162.4$ | $204.5$ | $169.8$ | $161.2$
Max Throughput [$\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{/}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{/}\mathrm{cycle}$] | $0.33$ | $0.23$ | $0.24$ | $0.25$
Avg. Latency _(Zero-load)_ [$\mathrm{cycle}$] | $4.7$ | $6.4$ | $7.9$ | $9.3$
Peak Performance _(Typ.)_ [$\mathrm{TOPS}$] | $0.47$ | $1.50$ | $1.80$ | $1.89$
Area Efficiency [$\mathrm{GOPS}\text{/}{\mathrm{mm}}^{2}$] | $47.0$ | $21.8$ | $26.1$ | $27.4$

* Subscript indicates core count; implemented in GF12LP+ technology.

##  4\. SDR Benchmarks Performance

TeraPool-SDR offers a streamlined fork-join programming model, highly suitable
for hardware-software co-design of SDR workloads. Targeting optimized
architecture-native kernel libraries, we adopt a low overhead C-runtime
approach and distribute portions of the input data to the PEs, according to
their ID (accessible via runtime primitives). After a parallel-section cores
are synchronized with a runtime barrier call. To keep utilization high and
avoid contentions to shared memory resources, PEs are constrained to fetch
data from portions of the shared SPMs they can access with low latency. Key
kernels for SDR workloads are implemented: FFT, beamforming MatMul, CHE, and
SysInv.

We adopt a radix-4 decimation in frequency Cooley-Turkey FFT, to ease the
memory accesses in local banks (Bertuletti et al., 2023). In the $k^{th}$
stage of an N-points FFT, each core computes $4$ butterflies, taking $4$
inputs at a distance of $N/(4\times 4k)$. Cores working on different FFTs are
independently synchronized.

The MatMul tiled implementation (Bertuletti et al., 2023), aims to fully
utilize register file in Snitch, maximizing computational intensity. The
implementation reduces interconnect stalls, through a cascaded
parallelization, where PEs of the same Tile shift the fetch address start
point along matrix rows, to avoid contentions for the same Group port.

In CHE, consisting of an element-wise matrix division, cores loop across
memory rows, compute the fetch-indexes via modulo operations, and only produce
the outputs residing in their local banks.

The SysInv leverages weak-scaling: as in the lower-PHY the per-subcarrier MIMO
has less than $32$ transceivers we assign an independent small squared SysInv
problem to each core and then synchronize. We use Cholesky Decomposition
(CholDec) to invert the linear system, and store the intermediate lower-
triangular decomposition in the local memory of the core working on it.

The large capacity of TeraPool-SDR’s L1 allows to set up functional pipelines
without resorting to L2 accesses to solve the data dependencies between the
different steps of lower-PHY processing (e.g., in a typical OFDM workload the
data of FFT for up to 64 antennas can be passed to the MatMul beamforming
stage with no L2 memory transfers).

Table 3. Performance metrics of SDR key-kernels.

|  | $\text{TP-SDR}_{\text{1-3-5-7}}$ | $\text{TP-SDR}_{\text{1-3-5-9}}$ | $\text{TP-SDR}_{\text{1-3-5-11}}$
---|---|---|---|---
FFT | Max Size in L1 |  64x4096 |  64x4096 |  64x4096
Total Cycles | $31\,740$ | $32\,012$ | $32\,272$
IPC | $0.75$ | $0.75$ | $0.74$
MatMul | Max Size in L1 |  512x512 |  512x512 |  512x512
Total Cycles | $296\,995$ | $298\,239$ | $301\,113$
IPC | $0.70$ | $0.69$ | $0.69$
CHE | Max Size in L1 | 4096 32x4 | 4096 32x4 | 4096 32x4
Total Cycles | $16\,851$ | $17\,221$ | $17\,601$
IPC | $0.66$ | $0.64$ | $0.63$
SysInv | Max Size in L1 | 65536 4x4 | 65536 4x4 | 65536 4x4
Total Cycles | $30\,539$ | $31\,254$ | $31\,870$
IPC | $0.61$ | $0.59$ | $0.58$

*Each core supports 8 outstanding transactions in the each given system.

Figure 5. Fraction of instructions and stalls over the total cycles for the
kernels execution in $\text{TeraPool-SDR}_{\text{1-3-5-7}}$.

Fraction of instructions and stalls over the total cycles for the kernels
execution in TeraPool-SDR 1-3-5-7

Table 3 demonstrates that TeraPool-SDR’s large shared-L1 memory enables
processing of the entire data frame in a typical 5G use case without
splitting, achieving an instructions-per-cycle (IPC) of up to 0.75 for key SDR
kernels. TeraPool-SDR’s remote Group access latency varies from 7 to 11
cycles, but the IPC loss across different configurations stays below $3$,
proving an effective latency-hiding in software and hardware. Figure 5
displays the detailed IPC and stall fractions for the execution of SDR kernels
on TeraPool-SDR. The parallelization scheme, minimizes Load Store Unit (LSU)
stalls by eliminating access to remote memory banks for FFT, CHE, and SysInv.
The read-after-write (RAW) and external-unit stalls in CHE and SysInv result
from multi-cycle instructions (mainly divisions) offloaded to pipelined
functional units during intensive computing loops. Being a control-heavy task,
SysInv experiences more I$ misses and synchronization overhead. For the
MatMul, although target data structures distribute over all the banks of the
shared SPM, causing unavoidable LSU stalls, the IPC on the TeraPool-SDR
Cluster still achieves $0.7$, highlighting that TeraPool-SDR delivers a
significant fraction of the peak workload even for kernels with PE-to-L1
traffic patterns that are not highly local, with non-negligible contention.

Figure 6 shows that TeraPool-SDR achieves comparable or higher energy
efficiency than a $4$$\times$ smaller-scale cluster on all the key SDR kernels
thanks to its low-power interconnect. $\text{TeraPool-SDR}_{\text{1-3-5-7}}$
is the lowest power consumption design: a memory request crossing the cluster
takes $13.5\text{\,}\mathrm{p}\mathrm{J}$, only 0.5$\times$ more than a local
request. The performance and energy efficiency of this design are however
dominated by the other two. $\text{TeraPool-SDR}_{\text{1-3-5-11}}$ is Pareto-
optimal in operating frequency and performance, at the cost of a higher power
consumption ($6.5\text{\,}\mathrm{W}$ for FFT, $8.8\text{\,}\mathrm{W}$ for
MatMul, $6.6\text{\,}\mathrm{W}$ for CHE and $4.9\text{\,}\mathrm{W}$ for
SysInv kernels). With minimal performance losses, $\text{TeraPool-
SDR}_{\text{1-3-5-9}}$ is optimal in energy efficiency for the selected SDR
workloads. Being the most compute-intensive task, MatMul achieves a peak of
$125\text{\,}\mathrm{GOPS}\text{/}\mathrm{W}$ and consumes less than
$6.4\text{\,}\mathrm{W}$, in compliance with the low power consumption
standards of base-station hardware.

Figure 6. Energy Efficiency for SDR workloads.

Energy Efficiency for SDR workloads, shows that TeraPool-SDR achieves
comparable or higher energy efficiency than a 4 times smaller-scale cluster on
all the key SDR kernels

##  5\. Conclusion

TeraPool-SDR is a physically feasible, many-core cluster of $1024$ Snitch
RISC-V cores sharing $4\text{\,}\mathrm{M}\mathrm{i}\mathrm{B}$ of L1 SPM,
through hierarchical low-latency NUMA TCDM interconnections (less than $5$
cycles within local Group, $7$-$11$ to remote Group, depending on the target
frequency). Completing PnR by using GlobalFoundries’ 12LPPLUS FinFET
technology, TeraPool-SDR occupies a die area of
$68.9\text{\,}\mathrm{m}\mathrm{m}^{2}$ and operates at up to
$924\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ ($63$ gate delays) in typical
operating conditions
(TT/$0.80\text{\,}\mathrm{V}$/$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$) with
a theoretical peak performance of $1.89\text{\,}\mathrm{TOPS}$.

TeraPool-SDR’s large PE count exceeds the state-of-the-art by $4$$\times$ and
achieves high performance. Depending on the target operating frequencies,
TeraPool-SDR achieves $0.18\text{\,}\mathrm{-}$$0.84\text{\,}\mathrm{TOPS}$
and $60\text{\,}\mathrm{-}$$125\text{\,}\mathrm{GOPS}\text{/}\mathrm{W}$ on
the benchmarks of 5G 7.X splits targeted for acceleration in base-station
hardware by industry-leading solutions. Thanks to its low-latency interconnect
with a large shared-L1, TeraPool-SDR minimizes the data splitting, transfer,
and synchronization overheads of memory-intensive SDR applications.

###### Acknowledgements.

This work is funded in part by the COREnext project supported by the EU
Horizon Europe research and innovation program under grant agreement No.
$101\,092\,598$.

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# Near-order relation of power means

Jinmi Hwang and Sejong Kim

###### Abstract.

On the setting of positive definite operators we study the near-order
properties of power means such as the quasi-arithmetic mean (Hölder mean) and
Rényi power mean. We see the monotonicity of spectral geometric mean and
Wasserstein mean on parameters with respect to the near-order and the near-
order relationship between the spectral geometric mean and Wasserstein mean.
Furthermore, the monotonicity of quasi-arithmetic mean on parameters and the
convergence of Rényi power mean to the log-Euclidean mean with respect to the
near-order have been established.

Mathematics Subject Classification (2020): 47B65, 15B48

Keywords: Near-order, metric geometric mean, spectral geometric mean,
Wasserstein mean, quasi-arithmetic mean, Rényi power mean

##  1\. Introduction

Kubo and Ando [19] have established the theory of operator means for two
variables on the open convex cone of positive definite operators. The crucial
property of Kubo-Ando’s operator mean is the monotonicity for Loewner partial
order:

$A\sigma B\leq C\sigma D\quad$ whenever $\quad A\leq C$ and $B\leq D$

for positive definite operators $A,B,C,D$, where $\sigma$ denotes the Kubo-
Ando’s operator mean. This is related to the operator monotone function
corresponding to the Kubo-Ando’s operator mean. On the other hand, there are
lots of two-variable means which do not fulfill the monotonicity such as the
spectral geometric mean [6, 21] and Wasserstein mean [2, 13].

One of the important multi-variable means of positive definite matrices is a
least squares mean for the Riemannian trace metric, named a Cartan (Karcher)
mean. It has been known in [16] that the Cartan mean coincides with a unique
positive definite solution $X$ of the Karcher equation

$\sum_{j=1}^{n}w_{j}\log(X^{1/2}A_{j}^{-1}X^{1/2})=0,$

where $A_{j}$’s are positive definite matrices and
$\omega=(w_{1},\dots,w_{n})$ is a positive probability vector. This allows us
to define the Karcher mean of positive definite operators [20]. The
monotonicity of Karcher mean is a long-standing open problem, but it has been
proved in [20] by using the approach of power means.

New order relation, called a _near-order_ , has been recently introduced in
[5]:

$A\preceq B\quad$ if and only if $\quad A^{-1}\\#B\geq I$.

Here, $A^{-1}\\#B=A^{-1/2}(A^{1/2}BA^{1/2})^{1/2}A^{-1/2}$ is the geometric
mean of $A^{-1}$ and $B$, which is a typical example of the Kubo-Ando’s
operator mean. One can easily see that $A\leq B$ implies $A\preceq B$. Unlike
the Loewner-Heinz inequality, an interesting property of the near-order is as
follows:

$A\preceq B\quad$ implies $\quad A^{p}\preceq B^{p}\quad$ for all $p\geq 1$.

One of the main results is the monotonicity of spectral geometric mean and
Wasserstein mean on parameters with respect to the near-order.

Several near-order inequalities of spectral geometric mean and Wasserstein
mean for two variables have been shown in [5, 8], but those of multi-variable
means are rare. In this paper we study two kinds of the multi-variable power
mean, named the quasi-arithmetic mean and Rényi power mean. For given positive
definite operators $A_{1},\dots,A_{n}$, the quasi-arithmetic mean $Q_{p}$ of
order $p\neq 0$ is defined by

$Q_{p}(\omega;A_{1},\dots,A_{n})=\left(\sum^{n}_{j=1}w_{j}A_{j}^{p}\right)^{1/p},$

and the Rényi power mean $\mathcal{R}_{t,z}$ for $0\leq t<z\leq 1$ is defined
as a unique positive definite solution $X$ of the equation

$X=\sum_{j=1}^{n}w_{j}\left(A_{j}^{\frac{1-t}{2z}}X^{\frac{t}{z}}A_{j}^{\frac{1-t}{2z}}\right)^{z}.$

The main results of multi-variable power means with respect to the near-order
are the monotonicity of quasi-arithmetic mean on both variables and parameters
and the inequalities of Rényi power mean with the quasi-arithmetic mean and
log-Euclidean mean. In particular, the monotonicity of quasi-arithmetic mean
provides the following chain: for $0<p\leq q\leq 1$

$Q_{-1/p}\leq Q_{-1/q}\leq Q_{-1}=\mathcal{H}\preceq Q_{-q}\preceq
Q_{-p}\preceq\mathrm{LE}\preceq Q_{p}\preceq Q_{q}\leq\mathcal{A}=Q_{1}\leq
Q_{1/q}\leq Q_{1/p},$

where $\displaystyle\mathrm{LE}=\lim_{p\to 0}Q_{p}$ denotes the log-Euclidean
mean.

As an extension of Lie-Trotter formula, new type of multi-variable mean,
called a Lie-Trotter mean, has been introduced in [12]. It is known that
multi-variable means between the arithmetic and harmonic means with respect to
the Loewner order are Lie-Trotter means, for instance, Karcher mean. We
generalize such a consequence to multi-variable means between the arithmetic
and harmonic means with respect to the near-order.

##  2\. Two-variable operator means

Let $B(\mathcal{H})$ be the Banach space of all bounded linear operators on a
Hilbert space $\mathcal{H}$ with inner product $\langle\cdot,\cdot\rangle$,
and let $S(\mathcal{H})\subset B(\mathcal{H})$ be the closed subspace of all
self-adjoint operators. We call $A\in S(\mathcal{H})$ positive semi-definite
if $\langle x,Ax\rangle\geq 0$ for all $x\in\mathcal{H}$, and positive
definite if $\langle x,Ax\rangle>0$ for all nonzero $x\in\mathcal{H}$. We
denote as $\mathbb{P}\subset S(\mathcal{H})$ the open convex cone of all
positive definite operators. The group GL of all invertible operators
transitively acts on $\mathbb{P}$ via congruence transformation. On the
finite-dimensional setting $\mathcal{H}=\mathbb{C}^{m}$ we write as
$\mathbb{H}_{m}$ and $\mathbb{P}_{m}$ respectively the real vector space of
all $m\times m$ Hermitian matrices and the open convex cone of all $m\times m$
positive definite matrices.

The Thompson metric $d_{T}$ on $\mathbb{P}$ is defined by

$d_{T}(A,B)=\max\\{\log M(A/B),\log M(B/A)\\},$

where $M(B/A)=\inf\\{\lambda>0:B\leq\lambda A\\}$. Here, $\leq$ denotes the
_Loewner partial order_ on $S(\mathcal{H})$. Note from [3, 22] that

$d_{T}(A,B)=\|\log A^{-1/2}BA^{-1/2}\|$

for the operator norm $\|\cdot\|$, and $d_{T}$ is a complete metric on
$\mathbb{P}$. The following are well-known fundamental properties for $d_{T}$.

######  Lemma 2.1.

Let $A,B,C,D\in\mathbb{P}$. The Thompson metric satisfies

  * (1)

$d_{T}(A,B)=d_{T}(A^{-1},B^{-1})=d_{T}(MAM^{*},MBM^{*})$ for any
$M\in\mathrm{GL}$;

  * (2)

$d_{T}(A+B,C+D)\leq\max\\{d_{T}(A,C),d_{T}(B,D)\\}$;

  * (3)

$d_{T}(A^{t},B^{t})\leq td_{T}(A,B)$ for any $t\in[0,1]$.

The _metric geometric mean_ of $A,B\in\mathbb{P}$ is defined as

$A\\#_{t}B:=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}$ (2.1)

for $t\in[0,1]$. We simply write $A\\#B=A\\#_{1/2}B$, and note that $A\\#B$ is
the unique solution $X\in\mathbb{P}$ of the Riccati equation $XA^{-1}X=B$. By
Lemma 2.1 the metric geometric mean satisfies the following convexity for
$d_{T}$:

$d_{T}(A\\#_{s}B,C\\#_{t}D)\leq(1-t)\,d_{T}(A,C)+t\,d_{T}(B,D)+|s-t|\,d_{T}(C,D)$
(2.2)

for $A,B,C,D\in\mathbb{P}$ and $s,t\in[0,1]$. This means that the map

$[0,1]\times\mathbb{P}^{2}\ni(t,A,B)\mapsto A\\#_{t}B\in\mathbb{P}$

is continuous for the Thompson metric. The definition (2.1) can be extended to
all $t\in\mathbb{R}$.

The _spectral geometric mean_ of $A,B\in\mathbb{P}$ for $t\in[0,1]$ is defined
as

$A\natural_{t}B=(A^{-1}\\#B)^{t}A(A^{-1}\\#B)^{t}.$ (2.3)

It is first introduced for $t=1/2$ by Fiedler and Pták [6], and many
properties analogous to the metric geometric mean have been studied [10, 18,
21]. Analogous to the metric geometric mean, the definition (2.3) can be
extended to all $t\in\mathbb{R}$. Especially from [9], the spectral geometric
mean is a geodesic for the semi-metric $d(A,B)=2\|\log(A^{-1}\\#B)\|$ on
$\mathbb{P}$:

$d(A\natural_{s}B,A\natural_{t}B)\leq|s-t|\,d(A,B)$

for any $s,t\in\mathbb{R}$.

The _Wasserstein mean_ of $A,B\in\mathbb{P}$ for $t\in[0,1]$ is defined as

$A\diamond_{t}B=(1-t)^{2}A+t^{2}B+t(1-t)\left[A(A^{-1}\\#B)+(A^{-1}\\#B)A\right].$

This can be written as

$A\diamond_{t}B=\left[I\nabla_{t}(A^{-1}\\#B)\right]A\left[I\nabla_{t}(A^{-1}\\#B)\right],$
(2.4)

where $A\nabla_{t}B:=(1-t)A+tB$ is the weighted arithmetic mean of $A$ and
$B$. Since the Wasserstein mean $A\diamond_{t}B$ is the congruence
transformation on $A\in\mathbb{P}$ via $I\nabla_{t}(A^{-1}\\#B)$, it can be
defined for all $t\in\mathbb{R}$. It is known from [13, Theorem 5.1] that
$A\diamond_{t}B$ is the unique solution $X\in\mathbb{P}$ of the equation

$I=(1-t)(A\\#X^{-1})+t(B\\#X^{-1}).$

In the finite-dimensional setting of $\mathbb{P}_{m}$, Wasserstein mean is the
least squares mean for Bures-Wasserstein distance:

$A\diamond_{t}B=\underset{X\in\mathbb{P}_{m}}{\arg\min}(1-t)d_{W}^{2}(X,A)+td_{W}^{2}(X,B),$

where
$d_{W}(A,B)=\left[\operatorname{tr}(A+B)-2\operatorname{tr}(A^{1/2}BA^{1/2})^{1/2}\right]^{1/2}$
is the Bures-Wasserstein distance of $A,B\in\mathbb{P}_{m}$. See [10, 13, 14]
for more properties of the Wasserstein mean.

##  3\. Near-order of positive definite operators

The metric geometric mean satisfies the monotonicity on variables with respect
to the Loewner order:

$A\leq B\ \text{and}\ C\leq D\quad\Longrightarrow\quad A\\#_{t}C\leq
B\\#_{t}D\quad\textrm{for all}\ t\in[0,1].$

By the monotonicity, if $A\leq B$ then $A^{-1}\\#B\geq I.$ From this point of
view, Dumitru and Franco [5] introduced a new relation $\preceq$ on
$\mathbb{P}$, which is called _near-order_. For $A,B\in\mathbb{P}$,

$A\preceq B\quad\text{if and only if}\quad A^{-1}\\#B\geq I.$ (3.5)

The reflexive, antisymmetric, and modified versions of transitive properties,
called the near-transitivity, hold for the near-order. Moreover, for
$A,B\in\mathbb{P}$

$A\leq B\quad\Longrightarrow\quad\log A\leq\log B\quad\Longrightarrow\quad
A\preceq B.$ (3.6)

Note that $A\ll B$, called the _chaotic order_ , if and only if $\log
A\leq\log B$. Some properties for chaotic order, and monotonicity and
convexity properties for near-order have been studied in [7] and [5]
respectively.

A map $f:S(\mathcal{H})\to S(\mathcal{H})$ is operator monotone (monotone
increasing) if

$f(A)\leq f(B)\quad$ whenever $\quad A\leq B$.

Note that the map $X\mapsto X^{p}$ for $p\in[0,1]$ on $\mathbb{P}$ is monotone
increasing and the map $X\mapsto X^{p}$ for $p\in[-1,0]$ is monotone
decreasing with respect to the Loewner order [1]. This is known as the
_Loewner-Heinz inequality_. On the other hand, the following holds for the
near-order.

######  Proposition 3.1.

[5] Let $A,B\in\mathbb{P}$. Then $A\preceq B$ implies

  * (i)

$A^{p}\preceq B^{p}\quad$ for $p\geq 1$, and

  * (ii)

$B^{p}\preceq A^{p}\quad$ for $p\leq-1$.

The in-betweenness property of the metric geometric mean for the Loewner order
is well-known:

$A\leq A\\#_{t}B\leq B\quad\text{whenever}\quad A\leq B$

for any $t\in[0,1]$. Unfortunately this is not satisfied for the spectral
geometric mean and Wasserstein mean. On the other hand, their in-betweenness
properties for the near-order have been shown:

$A\preceq A\natural_{t}B\preceq B\quad\textrm{and}\quad A\preceq
A\diamond_{t}B\preceq B.$

In addition, we summarize the equivalent relations for $A\preceq B$ in terms
of metric, spectral geometric means and Wasserstein mean.

######  Theorem 3.2.

[5, 8] Let $A,B\in\mathbb{P}$. The following are equivalent:

  * (1)

$A\preceq B$,

  * (2)

$A^{-1}\\#B\geq I$,

  * (3)

$A\\#B^{-1}\leq I$,

  * (4)

$A\natural_{t}B\succeq A$ for some $t\in(0,1]$,

  * (5)

$A\natural_{t}B\preceq B$ for some $t\in[0,1)$,

  * (6)

$A\diamond_{t}B\succeq A$ for some $t\in(0,1]$,

  * (7)

$A\diamond_{t}B\preceq B$ for some $t\in[0,1)$.

For any $s,t\in\mathbb{R}$ such that $s\leq t$, if $A\leq B$ then
$A^{-1/2}BA^{-1/2}\geq I$ so

$A\\#_{s}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{s}A^{1/2}\leq
A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}=A\\#_{t}B.$

That is, the geodesic curve $A\\#_{t}B$ is monotone increasing on
$t\in\mathbb{R}$ in terms of the Loewner order when $A\leq B$.

In [8, Theorem 3.6], characterizations of the monotonicity for the spectral
geometric and Wasserstein means on parameters with respect to the near-order
have been shown on $\mathbb{P}_{m}$ by applying Lemma 3.3. These can be
extended to the infinite-dimensional setting $\mathbb{P}$ and
$s,t\in\mathbb{R}$.

######  Lemma 3.3.

[8] Let $X,Y\in\mathbb{P}$ such that $X,Y$ commute. Then $X\leq Y$ if and only
if $XAX\preceq YAY$ for any $A\in\mathbb{P}$.

######  Theorem 3.4.

Let $A,B\in\mathbb{P}$ and $s,t\in\mathbb{R}$ such that $s<t$. Then the
following are equivalent:

  * (i)

$A\preceq B$,

  * (ii)

$A\natural_{s}B\preceq A\natural_{t}B$,

  * (iii)

$A\diamond_{s}B\preceq A\diamond_{t}B$.

###### Proof.

The equivalences among (i), (ii) and (iii) have been proved in [8, Theorem
3.6], except (i) $\Leftrightarrow$ (iii) for $s,t\in\mathbb{R}$ such that
$s<t$.

We prove that (iii) implies (i). When $s=0$ and $t=1$, the conclusion is
obviously true. So we need to prove it when $s\neq 0$ and $t\neq 1$.

Set $C=A^{-1}\\#B$. By assumption (iii),
$(I\nabla_{s}C)A(I\nabla_{s}C)\preceq(I\nabla_{t}C)A(I\nabla_{t}C)$ for
certain $s,t\in\mathbb{R}$ such that $s<t$ so

$I\leq\left[(I\nabla_{s}C)^{-1}A^{-1}(I\nabla_{s}C)^{-1}\right]\\#\left[(I\nabla_{t}C)A(I\nabla_{t}C)\right].$

Taking congruence transformations by $I\nabla_{s}C$ and $A^{1/2}$
consecutively yield

$\begin{split}A^{1/2}(I\nabla_{s}C)^{2}A^{1/2}&\leq
I\\#A^{1/2}(I\nabla_{s}C)(I\nabla_{t}C)A(I\nabla_{t}C)(I\nabla_{s}C)A^{1/2}\\\
&=I\\#(A^{1/2}(I\nabla_{s}C)(I\nabla_{t}C)A^{1/2})^{2}=A^{1/2}(I\nabla_{s}C)(I\nabla_{t}C)A^{1/2}.\end{split}$

The first equality follows from the fact that $I\nabla_{s}C$ and
$I\nabla_{t}C$ commute. Taking congruence transformation by $A^{-1/2}$ we
obtain $(I\nabla_{s}C)^{2}\leq(I\nabla_{s}C)(I\nabla_{t}C)$, and hence,
$I\nabla_{s}C\leq I\nabla_{t}C$. This is equivalent to $C\geq I$ because
$s<t$, which means $A\preceq B$. ∎

The following shows the monotonicity of spectral geometric and Wasserstein
means on variables with respect to the near-order.

######  Proposition 3.5.

Let $A,B,C\in\mathbb{P}$ such that
$(A^{1/2}BA^{1/2})^{1/2}A^{-1}(A^{1/2}CA^{1/2})^{1/2}$ is self-adjoint. Then
$B\leq C$ implies

  * (1)

$A\natural_{t}B\preceq A\natural_{t}C$ for any $t\in(0,1]$,

  * (2)

$A\natural_{t}B\succeq A\natural_{t}C$ for any $t\in[-1,0)$, and

  * (3)

$A\diamond_{t}B\preceq A\diamond_{t}C$ for any $t\in(0,1]$.

###### Proof.

Set $X=A^{-1}\\#B$ and $Y=A^{-1}\\#C$. Then $X\leq Y$ by the monotonicity of
metric geometric mean with $B\leq C$, and $X^{t}\leq Y^{t}$ for $0<t\leq 1$ by
the Loewner-Heinz inequality. By assumption,

$(A^{1/2}BA^{1/2})^{1/2}A^{-1}(A^{1/2}CA^{1/2})^{1/2}=(A^{1/2}CA^{1/2})^{1/2}A^{-1}(A^{1/2}BA^{1/2})^{1/2},$

and $XY=YX$. By Lemma 3.3

$A\natural_{t}B=X^{t}AX^{t}\preceq Y^{t}AY^{t}=A\natural_{t}C.$

Similarly, we can obtain (2) for $t=-s$ where $s\in(0,1]$. Moreover, $X\leq Y$
yields $I\nabla_{t}X\leq I\nabla_{t}Y$ for $t\in(0,1]$, and $XY=YX$ yields
that $I\nabla_{t}X,I\nabla_{t}Y$ commute. By Lemma 3.3

$A\diamond_{t}B=[I\nabla_{t}X]A[I\nabla_{t}X]\preceq[I\nabla_{t}Y]A[I\nabla_{t}Y]=A\diamond_{t}C.$

∎

Motivated from [10, Theorem 4.4], Gan and Huang [8] have shown the near-order
relation between the spectral geometric mean and Wasserstein mean of positive
definite matrices. It can be extended to the infinite-dimensional setting
$\mathbb{P}$:

$A\natural_{t}B\preceq A\diamond_{t}B$ (3.7)

for $A,B\in\mathbb{P}$ and $t\in[0,1]$. Since the near-order is not invariant
under congruence transformation, the following is new.

######  Corollary 3.6.

Let $A,B\in\mathbb{P}$ and $t\in[0,1]$. Then

  * (1)

$A\leq B$ implies $A^{-1}\diamond B\geq I$, and

  * (2)

$A^{1/2}(A\natural_{t}B)A^{1/2}\preceq A^{1/2}(A\diamond_{t}B)A^{1/2}$.

###### Proof.

Note from [14, Remark 3.5] that $A\leq B$ is equivalent to $A^{-1}\natural
B\geq I$. By (3.7) and applying the near-transitivity in [5], we obtain (1).

Since $(A^{-1}\\#B)^{t}\leq I\nabla_{t}(A^{-1}\\#B)$,

$A^{1/2}(A^{-1}\\#B)^{t}A^{1/2}\leq A^{1/2}[I\nabla_{t}(A^{-1}\\#B)]A^{1/2},$

and $A^{1/2}(A^{-1}\\#B)^{t}A^{1/2}\preceq
A^{1/2}[I\nabla_{t}(A^{-1}\\#B)]A^{1/2}$. By Proposition 3.1 (i)

$\left[A^{1/2}(A^{-1}\\#B)^{t}A^{1/2}\right]^{2}\preceq\left[A^{1/2}[I\nabla_{t}(A^{-1}\\#B)]A^{1/2}\right]^{2},$

which completes the proof of (2). ∎

The operator fidelity of $A,B\in\mathbb{P}$ is given by

$F(A,B)=(A^{1/2}BA^{1/2})^{1/2}.$

Indeed, $\operatorname{tr}F(A,B)$ for density matrices $A,B$ is known as the
quantum fidelity. In general, $F(A,B)\neq F(B,A)$ although
$\operatorname{tr}F(A,B)=\operatorname{tr}F(B,A)$.

######  Proposition 3.7.

Let $A,B\in\mathbb{P}$. Then for all $n\in\mathbb{N}$

  * (1)

$A^{1/2}\preceq F(A,B)$ implies $A^{2^{n-1}}\preceq F(A^{2^{n}},B)$, and

  * (2)

$F(B,A)\preceq B^{1/2}$ implies $F(B^{2^{n}},A)\preceq B^{2^{n-1}}$.

###### Proof.

By Proposition 3.1 (i), the assumption of (1) implies $A\preceq
A^{1/2}BA^{1/2}$, and by definition of the near-order

$A^{-1}\\#(A^{1/2}BA^{1/2})\geq I.$

This is equivalent to $(ABA)^{1/2}\geq A$, and hence,
$A\preceq(ABA)^{1/2}=F(A^{2},B)$. So (1) holds for $n=1$.

Assume that $A^{2^{k-1}}\preceq F(A^{2^{k}},B)$ for $k\in\mathbb{N}$.
Replacing $A^{2^{k}}$ by $C$, the assumption can be written as $C^{1/2}\preceq
F(C,B)$. By the preceding argument for $n=1$ we get $C\preceq F(C^{2},B)$,
that is, $A^{2^{k}}\preceq F(A^{2^{k+1}},B)$. By induction on $n$, we proved
(1). ∎

######  Remark 3.8.

We recall the _entrywise eigenvalue relation_ $\leq_{\lambda}$ on
$\mathbb{H}_{m}$. Let
$\lambda_{1}(A)\geq\lambda_{2}(A)\geq\cdots\geq\lambda_{m}(A)$, where
$\lambda_{i}(A)$’s are real eigenvalues of $A\in\mathbb{H}_{m}$. For
$A,B\in\mathbb{H}_{m}$ we write as

$A\leq_{\lambda}B\quad\text{if and only
if}\quad\lambda_{i}(A)\leq\lambda_{i}(B)\ \textrm{for all}\ 1\leq i\leq m.$
(3.8)

This relation $\leq_{\lambda}$ is reflexive and transitive, but not
antisymmetric. We write as $A=_{\lambda}B$ if and only if
$\lambda_{i}(A)=\lambda_{i}(B)$ for all $i$. Note from [8] that for
$A,B\in\mathbb{P}_{m}$

$A\leq B\ \Longrightarrow\ A\preceq B\ \Longrightarrow\ A\leq_{\lambda}B\
\Longrightarrow\ A\prec_{w\log}B,$ (3.9)

where $A\prec_{w\log}B$ denotes that $A$ is weakly log-majorized by $B$. By
definition of the weak log-majorization, one can easily see that
$A\prec_{w\log}B$ and $B\prec_{w\log}A$ imply $A=_{\lambda}B$.

  * (1)

For $A,B\in\mathbb{H}_{m}$, $A=_{\lambda}B$ if and only if $A,B$ are unitarily
similar. Indeed, since $A,B\in\mathbb{H}_{m}$, they are unitarily
diagonalizable, that is,

$A=U_{A}D_{A}U_{A}^{*}\quad$ and $\quad B=U_{B}D_{B}U_{B}^{*}$,

where $U_{A},U_{B}$ are unitary matrices and $D_{A},D_{B}$ are diagonal
matrices whose main diagonal entries are real eigenvalues of $A,B$
respectively. If $A=_{\lambda}B$ then there exists a permutation matrix $P$
such that $D_{B}=PD_{A}P^{*}$. Thus,

$B=U_{B}PD_{A}P^{*}U_{B}^{*}=U_{B}P(U_{A}^{*}AU_{A})P^{*}U_{B}^{*}=VAV^{*},$

where $V:=U_{B}PU_{A}^{*}$ is a unitary matrix. The converse is true from [11,
Corollary 1.3.4].

  * (2)

It is natural to ask what kinds of $A,B\in\mathbb{P}_{m}$ yield that the
reverse implications of (3.9) hold. For instance, assuming for
$A,B\in\mathbb{P}_{m}$ that $A\preceq B$ implies $A\leq B$, what can we say
about $A$ and $B$? The following are sufficient conditions to satisfy the
reverse implications of (3.9):

    * (a)

$A\leq I\leq B$,

    * (b)

$AB=BA$.

##  4\. Monotonicity of quasi-arithmetic mean with respect to the near-order

Let $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathbb{P}^{n}$, and
$\omega=(w_{1},\dots,w_{n})\in\Delta_{n}$, the simplex of all positive
probability vectors in $\mathbb{R}^{n}$. The _quasi-arithmetic mean_ of order
$p\in\mathbb{R}\backslash\\{0\\}$ is defined by

$Q_{p}(\omega;\mathbb{A})=\left(\sum^{n}_{j=1}w_{j}A_{j}^{p}\right)^{1/p}.$

It is also called the Hölder mean or Pythagorean mean. Note that for $p=1$ and
$p=-1$

$Q_{1}(\omega;\mathbb{A})=\sum^{n}_{j=1}w_{j}A_{j}=:\mathcal{A}(\omega;\mathbb{A})\quad\text{and}\quad
Q_{-1}(\omega;\mathbb{A})=\left(\sum^{n}_{j=1}w_{j}A_{j}^{-1}\right)^{-1}=:\mathcal{H}(\omega;\mathbb{A}),$

which are the arithmetic and harmonic means, respectively. Moreover, the
quasi-arithmetic mean converges to the log-Euclidean mean as $p$ goes to $0$:

$\lim_{p\rightarrow
0}Q_{p}(\omega;\mathbb{A})=\exp\left[\sum^{n}_{j=1}w_{j}\log
A_{j}\right]=:\mathrm{LE}(\omega;\mathbb{A}).$ (4.10)

Also, the quasi-arithmetic mean satisfies the duality:

$Q_{p}(\omega;\mathbb{A})=Q_{-p}(\omega;\mathbb{A}^{-1})^{-1}.$ (4.11)

For $p\geq 1,$ the quasi-arithmetic satisfies the following monotonicity on
parameters with respect to the Loewner order:

######  Theorem 4.1.

[17] Let $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathbb{P}^{n}$ and
$\omega=(w_{1},\dots,w_{n})\in\Delta_{n}$. For $1\leq s\leq t<\infty,$

$Q_{-t}(\omega;\mathbb{A})\leq
Q_{-s}(\omega;\mathbb{A})\leq\mathcal{H}(\omega,\mathbb{A})\leq\mathcal{A}(\omega,\mathbb{A})\leq
Q_{s}(\omega;\mathbb{A})\leq Q_{t}(\omega;\mathbb{A}).$

Unfortunately, the quasi-arithmetic mean is not monotone for $p\in(-1,1)$ with
respect to the Loewner order. Also, it is not monotone on variables, that is,
$Q_{p}(\omega;\mathbb{A})\nleq Q_{p}(\omega;\mathbb{B})$ if $A_{j}\leq B_{j}$
for all $j=1,\dots,n,$ where
$\mathbb{A}=(A_{1},\dots,A_{n}),\mathbb{B}=(B_{1},\dots,B_{n})\in\mathbb{P}^{n}$;
see [17] for more information.

On the other hand, we show that the monotonicity of the quasi-arithmetic mean
on variables with respect to the near-order.

######  Theorem 4.2.

Let
$\mathbb{A}=(A_{1},\dots,A_{n}),\mathbb{B}=(B_{1},\dots,B_{n})\in\mathbb{P}^{n}$
and $\omega=(w_{1},\dots,w_{n})\in\Delta_{n}$. If $A_{j}\leq B_{j}$ for all
$j=1,\dots,n$, then for $0<p\leq 1$ or $-1\leq p<0$

$Q_{p}(\omega;\mathbb{A})\preceq Q_{p}(\omega;\mathbb{B}).$ (4.12)

###### Proof.

We first prove (4.12) for $0<p\leq 1$. If $A_{j}\leq B_{j}$ for $j=1,\dots,n$,
then $A_{j}^{p}\leq B_{j}^{p}$ by the Loewner-Heinz inequality. It implies
that
$\displaystyle\sum^{n}_{j=1}w_{j}A_{j}^{p}\leq\sum^{n}_{j=1}w_{j}B_{j}^{p}$.
So we obtain

$\sum^{n}_{j=1}w_{j}A_{j}^{p}\preceq\sum^{n}_{j=1}w_{j}B_{j}^{p}.$

Since $\frac{1}{p}\geq 1$, applying Proposition 3.1 (i) to the above relation
yields

$Q_{p}(\omega;\mathbb{A})=\left(\sum^{n}_{j=1}w_{j}A_{j}^{p}\right)^{1/p}\preceq\left(\sum^{n}_{j=1}w_{j}B_{j}^{p}\right)^{1/p}=Q_{p}(\omega;\mathbb{B}).$

By the similar argument with Proposition 3.1 (ii), we can prove (4.12) for
$-1\leq p<0$. ∎

######  Lemma 4.3.

Let $f,g:\mathbb{R}\rightarrow\mathbb{P}$ with $f(x)\preceq g(x)$ for every
$x\in\mathbb{R}$. Then

$\lim_{x\rightarrow a}f(x)\preceq\lim_{x\rightarrow a}g(x).$

###### Proof.

By assumption, $f(x)^{-1}\\#g(x)\geq I$ for every $x\in\mathbb{R}$. Since the
metric geometric mean and inversion are continuous for the Thompson metric by
(2.2) and Lemma 2.1 (1),

$\left(\lim_{x\rightarrow a}f(x)\right)^{-1}\\#\lim_{x\rightarrow
a}g(x)=\lim_{x\rightarrow a}\left[f(x)^{-1}\\#g(x)\right]\geq I.$

Thus, we obtain the desired relation. ∎

The log-majorization relationship among the Cartan, log-Euclidean, and
Wasserstein mean has been shown [2]: for
$\mathbb{A}=(A_{1},\dots,A_{n})\in\mathbb{P}_{m}^{n}$ and
$\omega=(w_{1},\dots,w_{n})\in\Delta_{n}$

$\Lambda(\omega;\mathbb{A})\prec_{\log}\mathrm{LE}(\omega;\mathbb{A})\prec_{w\log}\Omega(\omega;\mathbb{A}).$
(4.13)

Since $\Omega(\omega;\mathbb{A})\leq\mathcal{A}(\omega;\mathbb{A})$, we obtain
from (3.9)

$\mathrm{LE}(\omega;\mathbb{A})\prec_{w\log}\mathcal{A}(\omega;\mathbb{A}).$
(4.14)

We do not provide only stronger relation between the log-Euclidean and
arithmetic means, but also prove the monotonicity of the quasi-arithmetic mean
for parameter with respect to the near-order.

######  Theorem 4.4.

Let $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathbb{P}^{n}$ and
$\omega=(w_{1},\dots,w_{n})\in\Delta_{n}$. For $0\leq p\leq q\leq 1,$

$\mathcal{H}(\omega;\mathbb{A})\preceq Q_{-q}(\omega;\mathbb{A})\preceq
Q_{-p}(\omega;\mathbb{A})\preceq\mathrm{LE}(\omega;\mathbb{A})\preceq
Q_{p}(\omega;\mathbb{A})\preceq
Q_{q}(\omega;\mathbb{A})\preceq\mathcal{A}(\omega;\mathbb{A}).$ (4.15)

###### Proof.

We prove (4.15) with the following steps.

  * (1)

We first show $Q_{q}(\omega;\mathbb{A})\preceq\mathcal{A}(\omega;\mathbb{A})$
for $0\leq q\leq 1$. Since the map $X\rightarrow X^{q}$ for $0\leq q\leq 1$ is
operator concave on $\mathbb{P}$, we get
$\displaystyle\sum^{n}_{j=1}w_{j}A_{j}^{q}\leq\left(\sum^{n}_{j=1}w_{j}A_{j}\right)^{p}$
so
$\displaystyle\sum^{n}_{j=1}w_{j}A_{j}^{q}\preceq\left(\sum^{n}_{j=1}w_{j}A_{j}\right)^{p}$.
Using Proposition 3.1 (i), we obtain

$Q_{q}(\omega;\mathbb{A})=\left(\sum^{n}_{j=1}w_{j}A_{j}^{q}\right)^{1/q}\preceq\sum^{n}_{j=1}w_{j}A_{j}=\mathcal{A}(\omega;\mathbb{A}).$

Similarly, we get
$\displaystyle\sum^{n}_{j=1}w_{j}A_{j}^{-q}\preceq\left(\sum^{n}_{j=1}w_{j}A_{j}^{-1}\right)^{q}$
by the concavity of the map $X\rightarrow X^{q}$ for $0\leq q\leq 1$. Applying
Proposition 3.1 (ii), we get

$\mathcal{H}(\omega;\mathbb{A})=\left(\sum^{n}_{j=1}w_{j}A^{-1}_{j}\right)^{-1}\preceq\left(\sum^{n}_{j=1}w_{j}A^{-q}_{j}\right)^{-1/q}=Q_{-q}(\omega;\mathbb{A}).$

  * (2)

We show $Q_{p}(\omega;\mathbb{A})\preceq Q_{q}(\omega;\mathbb{A})$ for $0\leq
p\leq q\leq 1$. Since $0<\frac{p}{q}\leq 1$, the previous result (1) yields
$\displaystyle\left(\sum^{n}_{j=1}w_{j}A_{j}^{p/q}\right)^{q/p}\preceq\sum^{n}_{j=1}w_{j}A_{j}$.
Replacing $A_{j}$ by $A_{j}^{q}$ we have

$\left(\sum^{n}_{j=1}w_{j}A_{j}^{p}\right)^{q/p}\preceq\sum^{n}_{j=1}w_{j}A_{j}^{q}.$

Since $\frac{1}{q}\geq 1$ we obtain the desired relation by Proposition 3.1
(i).

  * (3)

Now we prove $Q_{-q}(\omega;\mathbb{A})\preceq Q_{-p}(\omega;\mathbb{A})$ for
$0\leq p\leq q\leq 1$. Applying the duality of quasi-arithmetic mean (4.11) to
the above result (2) with $-1\leq-q\leq-p\leq 0$,

$Q_{-q}(\omega;\mathbb{A})=Q_{q}(\omega;\mathbb{A}^{-1})^{-1}\preceq
Q_{p}(\omega;\mathbb{A}^{-1})^{-1}=Q_{-p}(\omega;\mathbb{A}).$

  * (4)

Using Lemma 4.3 and the property (4.10), we obtain
$\mathrm{LE}(\omega;\mathbb{A})\preceq Q_{p}(\omega;\mathbb{A})$ and
$Q_{-p}(\omega;\mathbb{A})\preceq\mathrm{LE}(\omega;\mathbb{A})$.

∎

######  Corollary 4.5.

The quasi-arithmetic mean of order $p$ converges to the log-Euclidean mean as
$p\to 0$ with respect to the near-order, in the sense that for
$\mathbb{A}\in\mathbb{P}^{n}$ and $\omega\in\Delta_{n}$

$\begin{split}Q_{p}(\omega;\mathbb{A})&\searrow_{\succeq}\mathrm{LE}(\omega;\mathbb{A})\quad\textrm{as}\quad
p\to 0^{+},\\\
Q_{p}(\omega;\mathbb{A})&\nearrow_{\preceq}\mathrm{LE}(\omega;\mathbb{A})\quad\textrm{as}\quad
p\to 0^{-}.\end{split}$

######  Remark 4.6.

For $1\leq p\leq q$ the following chain holds:

$Q_{-q}\leq Q_{-p}\leq\mathcal{H}\preceq Q_{-1/p}\preceq
Q_{-1/q}\preceq\mathrm{LE}\preceq Q_{1/q}\preceq Q_{1/p}\leq\mathcal{A}\leq
Q_{p}\leq Q_{q}.$

##  5\. Boundedness of Rényi power mean with respect to the near-order

The Rényi power mean, introduced in [4], is defined as a unique solution
$X\in\mathbb{P}$ of the equation

$X=\sum_{j=1}^{n}w_{j}Q_{t,z}(A_{j},X)=\sum_{j=1}^{n}w_{j}\left(A_{j}^{\frac{1-t}{2z}}X^{\frac{t}{z}}A_{j}^{\frac{1-t}{2z}}\right)^{z}$
(5.16)

for $0\leq t<z\leq 1$. We denote it as $\mathcal{R}_{t,z}(\omega;\mathbb{A})$,
where $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathbb{P}^{n}$ and
$\omega=(w_{1},\dots,w_{n})\in\Delta_{n}$.

######  Remark 5.1.

The function $f:\mathbb{P}\to\mathbb{P}$ defined by

$f(X)=\sum_{j=1}^{n}w_{j}\left(A_{j}^{\frac{1-t}{2z}}X^{\frac{t}{z}}A_{j}^{\frac{1-t}{2z}}\right)^{z},\quad
X\in\mathbb{P}$

for $0\leq t<z\leq 1$ is monotone increasing. Moreover, the Banach fixed point
theorem yields that

$\lim_{k\to\infty}f^{k}(Z)=\mathcal{R}_{t,z}(\omega;\mathbb{A})\quad\textrm{for
any}\ Z\in\mathbb{P}.$

Let $p\in\mathbb{R}$, $S\in\mathrm{GL}$, and $\sigma$ a permutation on
$\\{1,\dots,n\\}$. For convenience, we denote

$\begin{split}\mathbb{A}^{p}&=(A_{1}^{p},\ldots,A_{n}^{p})\in\mathbb{P}^{n}\\\
S\mathbb{A}S^{*}&=(SA_{1}S^{*},\ldots,SA_{n}S^{*})\in\mathbb{P}^{n}\\\
\mathbb{A}_{\sigma}&=(A_{\sigma(1)},\ldots,A_{\sigma(n)})\in\mathbb{P}^{n}\\\
\omega_{\sigma}&=(w_{\sigma(1)},\ldots,w_{\sigma(n)})\in\Delta_{n},\end{split}$

and

$\begin{split}\mathbb{A}^{(k)}&=(\underbrace{A_{1},\ldots,A_{n}},\dots,\underbrace{A_{1},\ldots,A_{n}})\in\mathbb{P}^{nk}\\\
\omega^{(k)}&=\frac{1}{k}(\underbrace{w_{1},\ldots,w_{n}},\dots,\underbrace{w_{1},\ldots,w_{n}})\in\Delta_{nk},\end{split}$

whose number of blocks is $k\in\mathbb{N}$.

######  Proposition 5.2.

Let $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathbb{P}^{n}$ and
$\omega=(w_{1},\dots,w_{n})\in\Delta_{n}$. Then

  * (1)

$\displaystyle\mathcal{R}_{t,z}(\omega;\mathbb{A})=Q_{1-t}(\omega;\mathbb{A})\quad$
if $A_{j}$’s commute;

  * (2)

$\displaystyle\mathcal{R}_{t,z}(\omega;c\mathbb{A})=c\mathcal{R}_{t,z}(\omega;\mathbb{A})\quad$
for any $c>0$;

  * (3)

$\displaystyle\mathcal{R}_{t,z}(\omega_{\sigma};\mathbb{A}_{\sigma})=\mathcal{R}_{t,z}(\omega;\mathbb{A})\quad$
for any permutation $\sigma$ on $\\{1,\dots,n\\}$;

  * (4)

$\displaystyle\mathcal{R}_{t,z}(\omega^{(k)};\mathbb{A}^{(k)})=\mathcal{R}_{t,z}(\omega;\mathbb{A})\quad$
for any $k\in\mathbb{N}$;

  * (5)

$\displaystyle\mathcal{R}_{t,z}(\omega;U\mathbb{A}U^{*})=U\mathcal{R}_{t,z}(\omega;\mathbb{A})U^{*}\quad$
for any unitary operator $U$;

  * (6)

$\displaystyle\mathcal{R}_{t,z}(\omega;\mathbb{A}^{-1})\geq\mathcal{R}_{t,z}(\omega;\mathbb{A})^{-1}$;

  * (7)

$\displaystyle\|\mathcal{R}_{t,z}(\omega;\mathbb{A})\|\leq\sum_{j=1}^{n}w_{j}\|A_{j}\|\quad$
for $1/2\leq t<z\leq 1$.

###### Proof.

All items (1)-(5) and (7) are already proved in [4, Proposition 4].

(6) Let $X=\mathcal{R}_{t,z}(\omega;\mathbb{A}^{-1})^{-1}$. Then
$X^{-1}=\mathcal{R}_{t,z}(\omega;\mathbb{A}^{-1})$, and by the arithmetic-
harmonic mean inequality

$X=\left[\sum_{j=1}^{n}w_{j}\left(A_{j}^{\frac{1-t}{2z}}X^{\frac{t}{z}}A_{j}^{\frac{1-t}{2z}}\right)^{-z}\right]^{-1}\leq\sum_{j=1}^{n}w_{j}\left(A_{j}^{\frac{1-t}{2z}}X^{\frac{t}{z}}A_{j}^{\frac{1-t}{2z}}\right)^{z}=f(X).$

By Remark 5.1 $X\leq f(X)\leq\cdots\leq f^{k}(X)$ for all $k\geq 1$. Taking
limit as $k\to\infty$ yields $X\leq\mathcal{R}_{t,z}(\omega;\mathbb{A})$. ∎

######  Remark 5.3.

For $A_{1},\dots,A_{n}\in\mathbb{P}_{m}^{n}$

$\det\mathcal{R}_{t,z}(\omega;\mathbb{A})\geq\prod_{j=1}^{n}(\det
A_{j})^{w_{j}}.$ (5.17)

Indeed, by using the joint concavity of $\log\det:\mathbb{P}_{m}\to\mathbb{R}$
to (5.16) for $X=\mathcal{R}_{t,z}(\omega;\mathbb{A})$

$\begin{split}\log\det X&\geq
z\sum_{j=1}^{n}w_{j}\log\det\left(A_{j}^{\frac{1-t}{2z}}X^{\frac{t}{z}}A_{j}^{\frac{1-t}{2z}}\right)\\\
&=(1-t)\sum_{j=1}^{n}w_{j}\log\det A_{j}+t\log\det X.\end{split}$

Thus, we obtain the determinantal inequality (5.17). Furthermore, the equality
in (5.17) holds if and only if $A_{1}=\cdots=A_{n}$.

######  Theorem 5.4.

Let $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathbb{P}^{n}$. Then for $0\leq t<z\leq
1$

  * (i)

$\displaystyle\mathcal{R}_{t,z}(\omega;\mathbb{A})^{\frac{1}{1-t}}\preceq
Q_{1-t}(\omega;\mathbb{A})\quad$ when
$\mathcal{R}_{t,z}(\omega;\mathbb{A})\leq I$,

  * (ii)

$\displaystyle\mathcal{R}_{t,z}(\omega;\mathbb{A})^{\frac{1}{1-t}}\succeq
Q_{1-t}(\omega;\mathbb{A})\quad$ when
$\mathcal{R}_{t,z}(\omega;\mathbb{A})\geq I$.

###### Proof.

Assume that $X=\mathcal{R}_{t,z}(\omega;\mathbb{A})\leq I$ for $0\leq t<z\leq
1$. Since
$\left(A_{j}^{\frac{1-t}{2z}}X^{\frac{t}{z}}A_{j}^{\frac{1-t}{2z}}\right)^{z}\leq
A_{j}^{1-t}$ for all $j$, we have

$X=\sum_{j=1}^{n}w_{j}\left(A_{j}^{\frac{1-t}{2z}}X^{\frac{t}{z}}A_{j}^{\frac{1-t}{2z}}\right)^{z}\leq\sum_{j=1}^{n}w_{j}A_{j}^{1-t}$
(5.18)

and $\displaystyle X\preceq\sum_{j=1}^{n}w_{j}A_{j}^{1-t}$. Since
$\frac{1}{1-t}\geq 1$, taking the $1/(1-t)$-power on both sides and applying
Proposition 3.1 (i) yield
$\displaystyle\mathcal{R}_{t,z}(\omega;\mathbb{A})^{\frac{1}{1-t}}\preceq
Q_{1-t}(\omega;\mathbb{A})$.

By the similar argument we can prove (ii). ∎

######  Remark 5.5.

Theorem 5.4 together with Theorem 4.4 we obtain the near-order relation
between the Rényi power mean and log-Euclidean mean:

$\displaystyle\mathcal{R}_{t,z}(\omega;\mathbb{A})^{\frac{1}{1-t}}\succeq\mathrm{LE}(\omega;\mathbb{A})\quad$
when $\mathcal{R}_{t,z}(\omega;\mathbb{A})\geq I$.

######  Theorem 5.6.

Let $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathbb{P}^{n}$. Then for $0\leq t<z\leq
1$

$\lim_{p\to
0^{-}}\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})^{1/p}\preceq\lim_{p\to
0^{+}}\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})^{1/p},$ (5.19)

when the limits exist. In addition,

  * (i)

$\displaystyle\lim_{p\to
0^{+}}\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})^{1/p}\preceq\mathrm{LE}(\omega;\mathbb{A}^{1-t})\quad$
when $A_{j}\leq I$ for all $j$,

  * (ii)

$\displaystyle\lim_{p\to
0^{-}}\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})^{1/p}\succeq\mathrm{LE}(\omega;\mathbb{A}^{1-t})\quad$
when $A_{j}\geq I$ for all $j$.

###### Proof.

Replacing $A_{j}$ by $A_{j}^{p}$ in Proposition 5.2 (6), we have

$\mathcal{R}_{t,z}(\omega;\mathbb{A}^{-p})^{-1}\leq\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})$

for $0<p<1$. So
$\mathcal{R}_{t,z}(\omega;\mathbb{A}^{-p})^{-1}\preceq\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})$,
and hence, by Proposition 3.1 (i)

$\mathcal{R}_{t,z}(\omega;\mathbb{A}^{-p})^{-1/p}\preceq\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})^{1/p}.$

Taking limit as $p\to 0^{+}$ we obtain (5.19) when the limits exist.

We first show

$\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})^{1/p}\preceq
Q_{p}(\omega;\mathbb{A}^{1-t})$ (5.20)

for $0<p<1$ when $A_{j}\leq I$ for all $j$. Let
$X=\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})$. Since $A_{j}^{p}\leq I$ for all
$j$, $X\leq I$ by [15, Lemma 5.2]. Then from (5.18) in the proof of Theorem
5.4 (i)

$\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})\leq\sum_{j=1}^{n}w_{j}A_{j}^{(1-t)p}$

So
$\displaystyle\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})\preceq\sum_{j=1}^{n}w_{j}A_{j}^{(1-t)p}$,
and we obtain (5.20) by Proposition 3.1 (i). Taking limit as $p\to 0^{+}$ and
applying (4.10) and Lemma 4.3, we complete the proof of (i).

We can prove

$\mathcal{R}_{t,z}(\omega;\mathbb{A}^{p})^{1/p}\succeq
Q_{p}(\omega;\mathbb{A}^{1-t})$

for $-1<p<0$ by the similar argument as above with Proposition 3.1 (ii).
Taking limit as $p\to 0^{-}$ and applying (4.10) and Lemma 4.3 we obtain (ii).
∎

##  6\. Final remarks and open question

By Theorem 4.4 we have already proved the weighted
arithmetic-$\mathrm{LE}$-harmonic mean inequalities for near-order:

$\mathcal{H}(\omega;\mathbb{A})\preceq\mathrm{LE}(\omega;\mathbb{A})\preceq\mathcal{A}(\omega;\mathbb{A})$
(6.21)

for $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathbb{P}^{n}$ and
$\omega=(w_{1},\dots,w_{n})\in\Delta_{n}$. On the other hand, we can simply
prove (6.21) by using operator concavity of the logarithmic map
$\log:\mathbb{P}\to S(\mathcal{H})$. Indeed, since

$\sum_{j=1}^{n}w_{j}\log A_{j}\leq\log\left(\sum_{j=1}^{n}w_{j}A_{j}\right)$

and the left-hand side can be written as $\displaystyle\sum_{j=1}^{n}w_{j}\log
A_{j}=\log\left(\mathrm{LE}(\omega;\mathbb{A})\right)$, we obtain the second
inequality of (6.21) by (3.6). Replacing $A_{j}$ by $A_{j}^{-1}$ we get the
first inequality of (6.21).

A weighted $n$-mean ($n\geq 2$) is a map
$G_{n}:=G_{n}(\omega;\cdot):\mathbb{P}^{n}\to\mathbb{P}$ satisfying the
idempotency, which means that $G_{n}(\omega;X,\dots,X)=X$ for all
$X\in\mathbb{P}$. The multivariable Lie-Trotter mean is the weighted mean
$G_{n}$ if it is differentiable and satisfies

$\lim_{s\to
0}G_{n}(\omega;\gamma_{1}(s),\dots,\gamma_{n}(s))^{1/s}=\mathrm{LE}(\omega;\gamma_{1}^{\prime}(0),\dots,\gamma_{n}^{\prime}(0)),$
(6.22)

where for $\epsilon>0$, $\gamma_{i}:(-\epsilon,\epsilon)\to\mathbb{P}$ are any
differentiable curves with $\gamma_{i}(0)=I$ for all $i$. By [12, Theorem 4.2]
if the weighted $n$-mean $G_{n}$ satisfies the weighted
arithmetic-$G_{n}$-harmonic mean inequalities for Loewner order:

$\mathcal{H}\leq G_{n}\leq\mathcal{A},$

then $G_{n}$ is the multivariable Lie-Trotter mean. We can provide the similar
result for near-order, but it is a generalization of [12, Theorem 4.2].

######  Theorem 6.1.

Let the weighted $n$-mean $G_{n}$ satisfy the weighted
arithmetic-$G_{n}$-harmonic mean inequalities for near-order:

$\mathcal{H}\preceq G_{n}\preceq\mathcal{A}.$ (6.23)

Then $G_{n}$ is the multivariable Lie-Trotter mean.

###### Proof.

Since the proof of [12, Lemma 4.1] follows for near-order, the weighted
$n$-mean $G_{n}$ is differentiable. It is enough to show (6.22). Let
$\gamma_{i}:(-\epsilon,\epsilon)\to\mathbb{P}$ be any differentiable curves
with $\gamma_{i}(0)=I$ for all $i$, where $0<\epsilon<1$. Then (6.23) implies

$\left[\sum_{j=1}^{n}w_{j}\gamma_{j}(s)^{-1}\right]^{-1}\preceq
G_{n}(\omega;\gamma_{1}(s),\dots,\gamma_{n}(s))\preceq\sum_{j=1}^{n}w_{j}\gamma_{j}(s).$

For $0<s<1$ we have from Proposition 3.1 (i)

$\left[\sum_{j=1}^{n}w_{j}\gamma_{j}(s)^{-1}\right]^{-1/s}\preceq
G_{n}(\omega;\gamma_{1}(s),\dots,\gamma_{n}(s))^{1/s}\preceq\left[\sum_{j=1}^{n}w_{j}\gamma_{j}(s)\right]^{1/s}.$

Since the weighted arithmetic and harmonic means are the multivariable Lie-
Trotter means by [12, Theorem 3.1], taking limit as $s\to 0^{+}$ and applying
Lemma 4.3 yield

$\mathrm{LE}(\omega;\gamma_{1}^{\prime}(0),\dots,\gamma_{n}^{\prime}(0))\preceq\lim_{s\to
0^{+}}G_{n}(\omega;\gamma_{1}(s),\dots,\gamma_{n}(s))^{1/s}\preceq\mathrm{LE}(\omega;\gamma_{1}^{\prime}(0),\dots,\gamma_{n}^{\prime}(0)).$

Since the near-order satisfies the antisymmetric property, we obtain (6.22).
For $-1<s<0$ we can get (6.22) by similar arguments. ∎

One can naturally ask whether Theorem 6.1 holds even though the condition
(6.23) is replaced by the entrywise eigenvalue relation or weak log-
majorization. Unfortunately, it is not true because the entrywise eigenvalue
relation and weak log-majorization do not satisfy the antisymmetric property.

######  Corollary 6.2.

Let the weighted $n$-mean $G_{n}$ satisfy the weighted
arithmetic-$G_{n}$-harmonic mean inequalities for weak log-majorization:

$\mathcal{H}\prec_{w\log}G_{n}\prec_{w\log}\mathcal{A}.$

If the limit $\displaystyle\lim_{s\to
0}G_{n}(\omega;\gamma_{1}(s),\dots,\gamma_{n}(s))^{1/s}$ exists for
differentiable curves $\gamma_{i}:(-\epsilon,\epsilon)\to\mathbb{P}_{m}$ with
$\gamma_{i}(0)=I$ for all $i$, then

$\lim_{s\to
0}G_{n}(\omega;\gamma_{1}(s),\dots,\gamma_{n}(s))^{1/s}=_{\lambda}\mathrm{LE}(\omega;\gamma_{1}^{\prime}(0),\dots,\gamma_{n}^{\prime}(0)).$

###### Proof.

Following the proof of Theorem 6.1 we obtain

$\mathrm{LE}(\omega;\gamma_{1}^{\prime}(0),\dots,\gamma_{n}^{\prime}(0))\prec_{w\log}\lim_{s\to
0^{+}}G_{n}(\omega;\gamma_{1}(s),\dots,\gamma_{n}(s))^{1/s}\prec_{w\log}\mathrm{LE}(\omega;\gamma_{1}^{\prime}(0),\dots,\gamma_{n}^{\prime}(0))$

for $0<s<1$. So $\displaystyle\lim_{s\to
0^{+}}G_{n}(\omega;\gamma_{1}(s),\dots,\gamma_{n}(s))^{1/s}=_{\lambda}\mathrm{LE}(\omega;\gamma_{1}^{\prime}(0),\dots,\gamma_{n}^{\prime}(0))$
when the limit exists. Similarly for $-1<s<0$, we can obtain
$\displaystyle\lim_{s\to
0^{-}}G_{n}(\omega;\gamma_{1}(s),\dots,\gamma_{n}(s))^{1/s}=_{\lambda}\mathrm{LE}(\omega;\gamma_{1}^{\prime}(0),\dots,\gamma_{n}^{\prime}(0))$.
∎

It would be an interesting problem to study whether the weak log-majorization
relation generalizes to the near-order or entrywise eigenvalue relation. Note
that (6.21) is a generalization of (4.14). From this point of view, we give an
open question:

$\mathrm{LE}(\omega;\mathbb{A})\preceq\Omega(\omega;\mathbb{A}).$

Acknowledgement

All authors contributed equally, and there is no conflict of interest. The
work of S. Kim was supported by the National Research Foundation of Korea
grant funded by the Korea government (MSIT) (No. NRF-2022R1A2C4001306). The
work of J. Hwang was supported by Basic Science Research Program through the
National Research Foundation of Korea funded by the Ministry of Education (No.
NRF-2022R1I1A1A01068411).

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